ID
stringlengths 8
10
| split
stringclasses 1
value | knowledge concept
stringlengths 8
177
| question
stringlengths 31
488
| option
stringlengths 38
473
| answer
stringclasses 5
values | image_path
imagewidth (px) 66
2.87k
| key
stringclasses 7
values | question number
int64 1
1.74k
| knowledge concept description
stringlengths 96
3.56k
|
|---|---|---|---|---|---|---|---|---|---|
3stpes_1 | testmini | Properties and Understanding of Squares | As shown in the figure, a circle is drawn with vertex C of the square as the center. What is the measure of the central angle ∠ECF? ( )° | A. 45; B. 60; C. 72; D. 90; E. No correct answer | D | 3steps_1 | 1,081 | Properties and Understanding of Squares:
1. A square is a special type of parallelogram. A parallelogram with one pair of adjacent sides equal and one right angle is called a square, also known as a regular quadrilateral.
2. Both pairs of opposite sides are parallel; all four sides are equal; adjacent sides are perpendicular to each other.
3. All four angles are 90°, and the sum of the interior angles is 360°.
4. The diagonals are perpendicular to each other; the diagonals are equal in length and bisect each other; each diagonal bisects a pair of opposite angles.
5. A square is both a centrally symmetric figure and an axisymmetric figure (with four lines of symmetry). |
|
3stpes_1 | testmini | Circumference of Circles | As shown in the figure, a circle is drawn with vertex C of the square as the center, and the radius of the circle is as shown in the figure. The circumference of this circle is () cm. (Use π = 3.14) | A. 50.24; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | B | 3steps_2 | 1,246 | Circumference of Circles:
1. A line segment passing through the center of a circle with both endpoints on the circle is called the diameter, usually denoted by the letter d. A line segment connecting the center of the circle to any point on the circle is called the radius, usually denoted by the letter r.
2. The circumference of a circle = diameter × pi (C = πd) or the circumference of a circle = 2 × radius × pi (C = 2πr).
3. The circumference of a semicircle can be calculated using the formula C = πr + 2r.
4. In the same circle or congruent circles, there are an infinite number of radii and an infinite number of diameters. All radii and diameters in the same circle are equal. The length of the diameter is twice the length of the radius, and the length of the radius is half the length of the diameter. |
|
3stpes_1 | testmini | Understanding Sectors | As shown in the figure, a circle is drawn with the vertex C of a square as the center. The circumference of the circle is 25.12 cm. The length of the arc EF corresponding to the central angle ∠ECF is () cm. | A. 50.24; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | D | 3steps_3 | 1,411 | Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle". |
|
3stpes_1 | testmini | Properties and Understanding of Squares;Circumference of Circles;Understanding Sectors | As shown in the figure, a circle is drawn with the vertex C of a square as the center. The circle intersects the sides BC and CD of the square at points E and F, respectively. What is the arc length of EF on this circle? ( ) cm.(π = 3.14) | A. 50.24; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | D | 3steps_multi | 1,576 | Properties and Understanding of Squares:
1. A square is a special type of parallelogram. A parallelogram with one pair of adjacent sides equal and one right angle is called a square, also known as a regular quadrilateral.
2. Both pairs of opposite sides are parallel; all four sides are equal; adjacent sides are perpendicular to each other.
3. All four angles are 90°, and the sum of the interior angles is 360°.
4. The diagonals are perpendicular to each other; the diagonals are equal in length and bisect each other; each diagonal bisects a pair of opposite angles.
5. A square is both a centrally symmetric figure and an axisymmetric figure (with four lines of symmetry).
Circumference of Circles:
1. A line segment passing through the center of a circle with both endpoints on the circle is called the diameter, usually denoted by the letter d. A line segment connecting the center of the circle to any point on the circle is called the radius, usually denoted by the letter r.
2. The circumference of a circle = diameter × pi (C = πd) or the circumference of a circle = 2 × radius × pi (C = 2πr).
3. The circumference of a semicircle can be calculated using the formula C = πr + 2r.
4. In the same circle or congruent circles, there are an infinite number of radii and an infinite number of diameters. All radii and diameters in the same circle are equal. The length of the diameter is twice the length of the radius, and the length of the radius is half the length of the diameter.
Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle". |
|
3stpes_2 | testmini | Properties of Cones | Mike has a conical water container. Each time he fills the cone with water and then pours it all into a cylindrical storage container. He repeats this process 6 times. How much water does he pour in total? ( ) cm3 | A. 314; B. 628; C. 1256; D. 2512; E. No correct answer | B | 3steps_1 | 1,082 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
|
3stpes_2 | testmini | Volume and Capacity of Cylinders | As shown in the diagram, Mike uses a conical container to pour a total of 628 cm3 of water into a cylindrical water storage container, just enough to fill it. What is the height of the cylinder in cm? | A. 4; B. 8; C. 16; D. 20; E. No correct answer | B | 3steps_2 | 1,247 | Volume and Capacity of Cylinders:
1. The formula for calculating the volume of a cylinder: The volume of a cylinder = base area × height.
2. In terms of letters: If V represents the volume of the cylinder, where r is the radius of the base and h is the height of the cylinder, then the formula for the volume of the cylinder is: V = πr²h.
3. The capacity of a cylinder usually refers to the amount of space a cylindrical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
4. π is generally taken as 3.14. |
|
3stpes_2 | testmini | Conversion Rates and Calculations Between Length Units | The height of the cylindrical water storage container is as shown in the figure. Its height is equivalent to ( ) meters. | A. 0.8; B. 0.08; C. 16; D. 8; E. No correct answer | B | 3steps_3 | 1,412 | Conversion Rates and Calculations Between Length Units:
1. Conversion Rates between Length Units: 1 kilometer (km) = 1000 meters (m); 1 meter (m) = 10 decimeters (dm); 1 decimeter (dm) = 10 centimeters (cm); 1 centimeter (cm) = 10 millimeters (mm); 1 millimeter (mm) = 1000 nanometers (nm)
2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate.
3. To convert kilometers to meters: multiply by 1000.
4. To convert meters to decimeters: multiply by 10.
5. To convert decimeters to centimeters: multiply by 10.
6. To convert centimeters to millimeters: multiply by 10.
7. To convert millimeters to nanometers: multiply by 1000.
8. To convert nanometers to millimeters: divide by 1000.
9. To convert millimeters to centimeters: divide by 10.
10. To convert centimeters to decimeters: divide by 10.
11. To convert decimeters to meters: divide by 10.
12. To convert meters to kilometers: divide by 1000. |
|
3stpes_2 | testmini | Properties of Cones;Volume and Capacity of Cylinders;Conversion Rates and Calculations Between Length Units | As shown in the figure, there is a cylindrical water storage container. Mike wants to fill it with water. He uses a conical water container, filling it completely with water each time and then pouring it all into the cylindrical container. After pouring water 6 times, the cylindrical container is just full. What is the height of the cylinder? ( ) m | A. 0.8; B. 0.08; C. 16; D. 8; E. No correct answer | B | 3steps_multi | 1,577 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length.
Volume and Capacity of Cylinders:
1. The formula for calculating the volume of a cylinder: The volume of a cylinder = base area × height.
2. In terms of letters: If V represents the volume of the cylinder, where r is the radius of the base and h is the height of the cylinder, then the formula for the volume of the cylinder is: V = πr²h.
3. The capacity of a cylinder usually refers to the amount of space a cylindrical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
4. π is generally taken as 3.14.
Conversion Rates and Calculations Between Length Units:
1. Conversion Rates between Length Units: 1 kilometer (km) = 1000 meters (m); 1 meter (m) = 10 decimeters (dm); 1 decimeter (dm) = 10 centimeters (cm); 1 centimeter (cm) = 10 millimeters (mm); 1 millimeter (mm) = 1000 nanometers (nm)
2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate.
3. To convert kilometers to meters: multiply by 1000.
4. To convert meters to decimeters: multiply by 10.
5. To convert decimeters to centimeters: multiply by 10.
6. To convert centimeters to millimeters: multiply by 10.
7. To convert millimeters to nanometers: multiply by 1000.
8. To convert nanometers to millimeters: divide by 1000.
9. To convert millimeters to centimeters: divide by 10.
10. To convert centimeters to decimeters: divide by 10.
11. To convert decimeters to meters: divide by 10.
12. To convert meters to kilometers: divide by 1000. |
|
3stpes_3 | testmini | Properties and Understanding of Trapezoids | As shown in the figure, quadrilateral ABCD is a trapezoid, with the length of the lower base being twice the length of the upper base. What is the length of AD in cm? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_1 | 1,083 | Properties and Understanding of Trapezoids:
1. A trapezoid (or trapezium) is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called the bases of the trapezoid: the longer base is called the lower base, and the shorter base is called the upper base; the other two sides are called the legs; the perpendicular segment between the two bases is called the height of the trapezoid.
2. A trapezoid with one leg perpendicular to the bases is called a right trapezoid.
3. A trapezoid with both legs equal in length is called an isosceles trapezoid.
4. The height of a trapezoid is the distance between the upper base and the lower base. |
|
3stpes_3 | testmini | Rotation | As shown in the figure, triangle DEC is rotated counterclockwise around point D by a certain angle, and it is found that point C' coincides with point A. What is the length of CD? ( ) cm | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_2 | 1,248 | Rotation:
1. The rotation of a figure involves the movement of every point on the figure in the plane around a fixed point by a fixed angle. It can be described by three elements: the center of rotation, the direction of rotation, and the angle of rotation.
2. The distances from corresponding points to the center of rotation are equal.
3. The angle between the line segment connecting corresponding points and the center of rotation equals the angle of rotation.
4. The figures before and after rotation are congruent, meaning their size and shape remain unchanged.
5. The center of rotation is the unique fixed point. |
|
3stpes_3 | testmini | Perimeter of Trapezoids | As shown in the figure, the perimeter of the isosceles trapezoid ABCD is () cm. | A. 4; B. 6; C. 8; D. 10; E. No correct answer | D | 3steps_3 | 1,413 | Perimeter of Trapezoids:
1. In a trapezoid, the parallel sides are called the bases. The longer base is called the lower base, and the shorter base is called the upper base. The other two sides are called the legs. The perimeter of a trapezoid is the sum of the upper base, lower base, and the two legs. The formula for the perimeter is: upper base + lower base + leg + leg, denoted as L = a + b + c + d.
2. The formula for the perimeter of an isosceles trapezoid is: upper base + lower base + 2 legs, denoted as L = a + c + 2b. |
|
3stpes_3 | testmini | Properties and Understanding of Trapezoids;Rotation;Perimeter of Trapezoids | As shown in the figure, quadrilateral ABCD is an isosceles trapezoid, with the length of the lower base being twice the length of the upper base. DE is the height of the isosceles trapezoid. After rotating triangle DEC counterclockwise by a certain angle around point D, point C' coincides with point A. What is the perimeter of the isosceles trapezoid ABCD? ( ) cm | A. 4; B. 6; C. 8; D. 10; E. No correct answer | D | 3steps_multi | 1,578 | Properties and Understanding of Trapezoids:
1. A trapezoid (or trapezium) is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called the bases of the trapezoid: the longer base is called the lower base, and the shorter base is called the upper base; the other two sides are called the legs; the perpendicular segment between the two bases is called the height of the trapezoid.
2. A trapezoid with one leg perpendicular to the bases is called a right trapezoid.
3. A trapezoid with both legs equal in length is called an isosceles trapezoid.
4. The height of a trapezoid is the distance between the upper base and the lower base.
Rotation:
1. The rotation of a figure involves the movement of every point on the figure in the plane around a fixed point by a fixed angle. It can be described by three elements: the center of rotation, the direction of rotation, and the angle of rotation.
2. The distances from corresponding points to the center of rotation are equal.
3. The angle between the line segment connecting corresponding points and the center of rotation equals the angle of rotation.
4. The figures before and after rotation are congruent, meaning their size and shape remain unchanged.
5. The center of rotation is the unique fixed point.
Perimeter of Trapezoids:
1. In a trapezoid, the parallel sides are called the bases. The longer base is called the lower base, and the shorter base is called the upper base. The other two sides are called the legs. The perimeter of a trapezoid is the sum of the upper base, lower base, and the two legs. The formula for the perimeter is: upper base + lower base + leg + leg, denoted as L = a + b + c + d.
2. The formula for the perimeter of an isosceles trapezoid is: upper base + lower base + 2 legs, denoted as L = a + c + 2b. |
|
3stpes_4 | testmini | Understanding Sectors | As shown in the figure, the square DEOF is within a sector with a central angle of 90°. What is the length of OD in cm? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | D | 3steps_1 | 1,084 | Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle". |
|
3stpes_4 | testmini | Properties and Understanding of Squares | As shown in the figure, quadrilateral DEOF is a square, and the diagonal OD = 1 cm. Then EF = ( ) cm, and OD and EF are ( ) to each other. | A. 4, parallel; B. 1, parallel; C. 2, perpendicular; D. 1, perpendicular; E. No correct answer | D | 3steps_2 | 1,249 | Properties and Understanding of Squares:
1. A square is a special type of parallelogram. A parallelogram with one pair of adjacent sides equal and one right angle is called a square, also known as a regular quadrilateral.
2. Both pairs of opposite sides are parallel; all four sides are equal; adjacent sides are perpendicular to each other.
3. All four angles are 90°, and the sum of the interior angles is 360°.
4. The diagonals are perpendicular to each other; the diagonals are equal in length and bisect each other; each diagonal bisects a pair of opposite angles.
5. A square is both a centrally symmetric figure and an axisymmetric figure (with four lines of symmetry). |
|
3stpes_4 | testmini | Area of Squares | As shown in the figure, the diagonals of square DFEO are OD and EF. What is the area of square DFEO? ( ) cm² | A. 0.5; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_3 | 1,414 | Area of Squares:
1. The area of a square is equal to the square of its side length: S = a * a.
2. The area of a square is equal to the square of the length of its diagonal divided by two. |
|
3stpes_4 | testmini | Understanding Sectors;Properties and Understanding of Squares;Area of Squares | As shown in the figure, the square DEOF is within a sector with a central angle of 90°. What is the area of the square? ( ) cm² | A. 0.5; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_multi | 1,579 | Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle".
Properties and Understanding of Squares:
1. A square is a special type of parallelogram. A parallelogram with one pair of adjacent sides equal and one right angle is called a square, also known as a regular quadrilateral.
2. Both pairs of opposite sides are parallel; all four sides are equal; adjacent sides are perpendicular to each other.
3. All four angles are 90°, and the sum of the interior angles is 360°.
4. The diagonals are perpendicular to each other; the diagonals are equal in length and bisect each other; each diagonal bisects a pair of opposite angles.
5. A square is both a centrally symmetric figure and an axisymmetric figure (with four lines of symmetry).
Area of Squares:
1. The area of a square is equal to the square of its side length: S = a * a.
2. The area of a square is equal to the square of the length of its diagonal divided by two. |
|
3stpes_5 | testmini | Area of Trapezoids | Given that quadrilateral ABCD is a trapezoid as shown in the figure, with an area of 20 cm², what is the height BF of trapezoid ABCD in cm? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_1 | 1,085 | Area of Trapezoids:
1. Using the letters a and b to represent the upper base and the lower base of a trapezoid, and the letter h to represent the height of the trapezoid, the formula for the area of a trapezoid can be expressed as S = 1/2 (a + b) × h. |
|
3stpes_5 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, quadrilateral ABCE is a parallelogram, AB = 3 cm, then CE = ( ) cm | A. 4; B. 3; C. 2; D. 1; E. No correct answer | B | 3steps_2 | 1,250 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral. |
|
3stpes_5 | testmini | Area of Triangles | As shown in the figure, the area of triangle BCE is () cm². | A. 6; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_3 | 1,415 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_5 | testmini | Area of Trapezoids;Properties and Understanding of Parallelograms;Area of Triangles | As shown in the figure, quadrilateral ABCD is a trapezoid, and quadrilateral ABCE is a parallelogram. Given that the area of trapezoid ABCD is 20 cm², what is the area of triangle BCE in cm²? | A. 6; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_multi | 1,580 | Area of Trapezoids:
1. Using the letters a and b to represent the upper base and the lower base of a trapezoid, and the letter h to represent the height of the trapezoid, the formula for the area of a trapezoid can be expressed as S = 1/2 (a + b) × h.
Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral.
Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_6 | testmini | Cardinal Directions (East, South, West, North) | As shown in the figure, point B is directly north of point A, and point C is directly east of point B. What is the positional relationship between AB and BC? ( ) | A. Parallel; B. Perpendicular; C. Cannot be determined; D. No correct answer | B | 3steps_1 | 1,086 | Cardinal Directions (East, South, West, North):
1. Maps usually use "up" to represent north, "down" to represent south, "left" to represent west, and "right" to represent east.
2. South is opposite to north, and west is opposite to east; northwest is opposite to southeast, and northeast is opposite to southwest.
3. East, south, west, and north are arranged in a clockwise direction. |
|
3stpes_6 | testmini | Properties and Understanding of Triangles | As shown in the figure, AB is perpendicular to BC. Mike needs 5 meters to walk from A to B, and 5 meters to walk from B to C. What is the measure of ∠BAC? ( )° | A. 45; B. 60; C. 72; D. 90; E. No correct answer | A | 3steps_2 | 1,251 | Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. |
|
3stpes_6 | testmini | Southeast, Southwest, Northeast, Northwest Directions | As shown in the figure, Mike's initial position is at point A. He walks 5 meters north to reach point B, then walks 5 meters east to reach point C. In which direction is point C from point A? | A. North by west 45°; B. South by east 45°; C. North by east 45°; D. South by west 45°; E. No correct answer | C | 3steps_3 | 1,416 | Southeast, Southwest, Northeast, Northwest Directions:
1.Northeast lies 45 degrees north of due east, southeast lies 45 degrees south of due east, northwest lies 45 degrees west of due north, and southwest lies 45 degrees west of due south. |
|
3stpes_6 | testmini | Cardinal Directions (East, South, West, North);Properties and Understanding of Triangles;Southeast, Southwest, Northeast, Northwest Directions | As shown in the figure, Mike's initial position is at point A. He walks 5 meters north to reach point B, and then walks 5 meters east to reach point C. In which direction is point C from point A? | A. North by west 45°; B. South by east 45°; C. North by east 45°; D. South by west 45°; E. No correct answer | C | 3steps_multi | 1,581 | Cardinal Directions (East, South, West, North):
1. Maps usually use "up" to represent north, "down" to represent south, "left" to represent west, and "right" to represent east.
2. South is opposite to north, and west is opposite to east; northwest is opposite to southeast, and northeast is opposite to southwest.
3. East, south, west, and north are arranged in a clockwise direction.
Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.
Southeast, Southwest, Northeast, Northwest Directions:
1.Northeast lies 45 degrees north of due east, southeast lies 45 degrees south of due east, northwest lies 45 degrees west of due north, and southwest lies 45 degrees west of due south. |
|
3stpes_7 | testmini | Properties of Cones | As shown in the figure, triangle ABC is the cross-section of a cone cut along its middle. The height of the cone is as shown in the figure. What is the height of triangle ABC? ( ) cm | A. 4; B. 3; C. 2; D. 1; E. No correct answer | B | 3steps_1 | 1,087 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
|
3stpes_7 | testmini | Area of Triangles | As shown in the figure, the area of triangle ABC is 3 cm², and its height is equal to the height OA of the cone. What is the length of the base BC of the triangle? ( ) cm | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_2 | 1,252 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_7 | testmini | Volume and Capacity of Cones | As shown in the figure, the height of the cone is OA, and the diameter of the base is BC. What is the volume of the cone? ( ) cm³ | A. 3π; B. 1π; C. 2π; D. No correct answer | B | 3steps_3 | 1,417 | Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_7 | testmini | Properties of Cones;Area of Triangles;Volume and Capacity of Cones | If the cone shown in the figure is cut along the middle, the area of the triangle ABC at the cut surface is 3 cm², then what is the volume of the cone in cm³? | A. 3π; B. 1π; C. 2π; D. No correct answer | B | 3steps_multi | 1,582 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length.
Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined.
Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_8 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, quadrilateral ABCD is a trapezoid, with the lower base being three times the length of the upper base. When the upper base is extended to point E, it becomes a parallelogram, specifically parallelogram ABCE in the figure. What is the length of AD in cm? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | B | 3steps_1 | 1,088 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral. |
|
3stpes_8 | testmini | Area of Trapezoids | As shown in the figure, quadrilateral ABCD is a trapezoid with an area of 48 cm². Its lower base is three times the length of its upper base, and the length of AD is as shown in the figure. What is the height AF of the trapezoid? ( ) cm | A. 6; B. 7; C. 8; D. 9; E. No correct answer | C | 3steps_2 | 1,253 | Area of Trapezoids:
1. Using the letters a and b to represent the upper base and the lower base of a trapezoid, and the letter h to represent the height of the trapezoid, the formula for the area of a trapezoid can be expressed as S = 1/2 (a + b) × h. |
|
3stpes_8 | testmini | Area of Parallelograms | In parallelogram ABCE, the lengths of AF and AE are as shown in the figure. What is the area of parallelogram ABCE? ( ) cm² | A. 45; B. 60; C. 72; D. 90; E. No correct answer | C | 3steps_3 | 1,418 | Area of Parallelograms:
1. The area of a parallelogram is equal to the base times the height, S = ah. |
|
3stpes_8 | testmini | Properties and Understanding of Parallelograms;Area of Trapezoids;Area of Parallelograms | As shown in the figure, quadrilateral ABCD is a trapezoid with an area of 48 cm². Its lower base is three times the length of its upper base. When the upper base is extended to point E, it becomes a parallelogram, specifically parallelogram ABCE in the figure. What is the area of parallelogram ABCE in cm²? | A. 45; B. 60; C. 72; D. 90; E. No correct answer | C | 3steps_multi | 1,583 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral.
Area of Trapezoids:
1. Using the letters a and b to represent the upper base and the lower base of a trapezoid, and the letter h to represent the height of the trapezoid, the formula for the area of a trapezoid can be expressed as S = 1/2 (a + b) × h.
Area of Parallelograms:
1. The area of a parallelogram is equal to the base times the height, S = ah. |
|
3stpes_9 | testmini | Understanding Circles | As shown in the figure, circles A and B have equal radii, both equal to 3 cm. A and B are the centers of the two circles, point A is on circle B, point B is on circle A, and point P is the intersection point of the two circles. Which of the following relationships between the lengths PA, PB, and AB is correct? | A. PA = PB = AB; B. PA > PB > AB; C. PA < PB < AB; D. PA = PB < AB; E. No correct answer | A | 3steps_1 | 1,089 | Understanding Circles:
1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point.
2. A circle is an axisymmetric figure.
3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r.
4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius.
5. All radii of a circle are equal in length, and all diameters are equal in length.
6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle. |
|
3stpes_9 | testmini | Properties and Understanding of Triangles | In the given triangle PAB, the lengths of PA, PB, and AB are as shown in the diagram. What is the measure of ∠PBA? ( )° | A. 45; B. 60; C. 72; D. 90; E. No correct answer | B | 3steps_2 | 1,254 | Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. |
|
3stpes_9 | testmini | Understanding Sectors | As shown in the figure, the radius of circle B is marked, and point A is on circle B. A and B are the centers of two circles, and point P is the intersection point of the two circles. At this time, the arc length of segment PA is ( ) cm. | A. 3π; B. 1π; C. 2π; D. No correct answer | B | 3steps_3 | 1,419 | Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle". |
|
3stpes_9 | testmini | Understanding Circles;Properties and Understanding of Triangles;Understanding Sectors | As shown in the figure, circles A and B have equal radii. Point A is on circle B, and point B is on circle A. A and B are the centers of the two circles, respectively. Point P is the intersection point of the two circles. Connect PA and PB. At this time, the arc length corresponding to segment PA is () cm. | A. 3π; B. 1π; C. 2π; D. No correct answer | B | 3steps_multi | 1,584 | Understanding Circles:
1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point.
2. A circle is an axisymmetric figure.
3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r.
4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius.
5. All radii of a circle are equal in length, and all diameters are equal in length.
6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle.
Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.
Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle". |
|
3stpes_10 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, quadrilateral ABCD is a parallelogram. What is the relationship between the lengths of AB and CD? | A. AB = CD; B. AB > CD; C. AB < CD; D. No correct answer | A | 3steps_1 | 1,090 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral. |
|
3stpes_10 | testmini | Understanding Circles | As shown in the figure, points A, B, and D are exactly on the circle with center C. Then the lengths of segments BC, AC, and CD satisfy ( ) | A. BC = AC = CD; B. BC > AC > CD; C. BC < AC < CD; D. BC = AC < CD; E. No correct answer | A | 3steps_2 | 1,255 | Understanding Circles:
1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point.
2. A circle is an axisymmetric figure.
3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r.
4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius.
5. All radii of a circle are equal in length, and all diameters are equal in length.
6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle. |
|
3stpes_10 | testmini | Properties and Understanding of Triangles | As shown in the figure, a circle is drawn with center C, and points A and B are exactly on the circle. The lengths of BC, AC, and CD are as shown in the figure, and AB = CD. What is the measure of ∠ABC? | A. 45°; B. 60°; C. 72°; D. 90°; E. No correct answer | B | 3steps_3 | 1,420 | Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. |
|
3stpes_10 | testmini | Properties and Understanding of Parallelograms;Understanding Circles;Properties and Understanding of Triangles | As shown in the figure, quadrilateral ABCD is a parallelogram. A circle with a radius of 1 cm is drawn with center C, and the other three vertices of the parallelogram ABCD are exactly on the circle. What is the measure of ∠ABC? | A. 45°; B. 60°; C. 72°; D. 90°; E. No correct answer | B | 3steps_multi | 1,585 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral.
Understanding Circles:
1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point.
2. A circle is an axisymmetric figure.
3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r.
4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius.
5. All radii of a circle are equal in length, and all diameters are equal in length.
6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle.
Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. |
|
3stpes_11 | testmini | Area of Rectangles | In the quadrilateral ABCD shown in the figure, which is a rectangle with an area of 20 cm², what is the length of BA in cm? | A. 4; B. 3; C. 2; D. 5; E. No correct answer | A | 3steps_1 | 1,091 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
|
3stpes_11 | testmini | Understanding Sectors | As shown in the figure, in rectangle ABCD, a sector with center B and radius BA intersects BC at point E. The square FBHG is inside the sector. What is the length of BG in cm? | A. 4; B. 3; C. 2; D. 5; E. No correct answer | A | 3steps_2 | 1,256 | Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle". |
|
3stpes_11 | testmini | Area of Squares | Given the length of the diagonal BG of square FBHG, what is the area of square FBHG? ( ) cm² | A. 16; B. 8; C. 6; D. 4; E. No correct answer | B | 3steps_3 | 1,421 | Area of Squares:
1. The area of a square is equal to the square of its side length: S = a * a.
2. The area of a square is equal to the square of the length of its diagonal divided by two. |
|
3stpes_11 | testmini | Area of Rectangles;Understanding Sectors;Area of Squares | As shown in the figure, quadrilateral ABCD is a rectangle with an area of 20 cm². Taking B as the center and BA as the radius, the sector intersects the length BC of the rectangle at point E. A square FBHG is drawn inside the sector. The area of square FBHG is () cm². | A. 16; B. 8; C. 6; D. 4; E. No correct answer | B | 3steps_multi | 1,586 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab.
Understanding Sectors:
1. A sector is a shape formed by a circular arc and the two radii connecting the endpoints of the arc to the center of the circle.
2. All radii in a sector are equal in length.
3. The part of the circle between two points A and B is called an "arc".
4. An angle with its vertex at the center of the circle is called a "central angle".
Area of Squares:
1. The area of a square is equal to the square of its side length: S = a * a.
2. The area of a square is equal to the square of the length of its diagonal divided by two. |
|
3stpes_12 | testmini | Area of Rectangles | The perimeter of rectangle ABCD is 20 cm. What is the length of AB in cm? | A. 4; B. 3; C. 2; D. 5; E. No correct answer | A | 3steps_1 | 1,092 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
|
3stpes_12 | testmini | Properties of Cylinders | As shown in the figure, the rectangle ABCD is a front view of a cylinder. The length and width are marked in the figure. What is the height of the cylinder ( ) cm, and what is the radius of the base ( ) cm? | A. 6, 4; B. 6, 3; C. 4, 2; D. 6, 2; E. No correct answer | D | 3steps_2 | 1,257 | Properties of Cylinders:
1. The top and bottom surfaces of a cylinder are called the bases.
2. A cylinder has a curved surface called the lateral surface.
3. The distance between the two bases of a cylinder is called the height.
4. A cylinder is formed by rotating a rectangle 180° around one of its edges.
5. The bases of a cylinder are circular.
6. The heights of all cylinders are equal. |
|
3stpes_12 | testmini | Surface Area of Cylinders | As shown in the figure, the surface area of the following cylinder is () cm². | A. 24π; B. 16π; C. 32π; D. No correct answer | C | 3steps_3 | 1,422 | Surface Area of Cylinders:
1. The lateral surface area of a cylinder = circumference of the base × height.
2. The area of the base of a cylinder = π × radius squared.
3. The surface area of a cylinder refers to the sum of its lateral surface area and the areas of its two bases. |
|
3stpes_12 | testmini | Area of Rectangles;Properties of Cylinders;Surface Area of Cylinders | As shown in the figure, the rectangle ABCD is the front view of a cylinder. The perimeter of rectangle ABCD is 20 cm. What is the surface area of this cylinder in cm²? | A. 24π; B. 16π; C. 32π; D. No correct answer | C | 3steps_multi | 1,587 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab.
Properties of Cylinders:
1. The top and bottom surfaces of a cylinder are called the bases.
2. A cylinder has a curved surface called the lateral surface.
3. The distance between the two bases of a cylinder is called the height.
4. A cylinder is formed by rotating a rectangle 180° around one of its edges.
5. The bases of a cylinder are circular.
6. The heights of all cylinders are equal.
Surface Area of Cylinders:
1. The lateral surface area of a cylinder = circumference of the base × height.
2. The area of the base of a cylinder = π × radius squared.
3. The surface area of a cylinder refers to the sum of its lateral surface area and the areas of its two bases. |
|
3stpes_13 | testmini | Area of Rectangles | As shown in the figure, the area of the rectangle is 12.56 cm². What is the length of AD (in cm)? | A. 50.24; B. 3.14; C. 12.56; D. 6.28; E. No correct answer | D | 3steps_1 | 1,093 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
|
3stpes_13 | testmini | Expanded View of Cylinders | The lateral surface of a cylinder is unfolded into a rectangle ABCD as shown in the figure. What is the diameter of the base of this cylinder ( ) cm, and what is its height ( ) cm?(Use π = 3.14) | A. 6.28, 1; B. 1, 2; C. 6.28, 2; D. 2, 2; E. No correct answer | D | 3steps_2 | 1,258 | Expanded View of Cylinders:
1. Formation of a Cylinder: A cylinder can be formed by rotating a rectangle around one of its edges, or by rolling a rectangle into a cylindrical shape.
2. Components of a Cylinder: A cylinder consists of bases and a lateral surface. The bases are two equal circular surfaces, referred to as the upper base and the lower base; the lateral surface is a curved surface that connects the upper and lower bases.
3. Unfolded Lateral Surface of a Cylinder: When the lateral surface of a cylinder is unfolded, it forms a rectangle or a square. If the unfolded surface is a rectangle, its length is the circumference of the circular base, and its width is the height of the cylinder; if the unfolded surface is a square, it means the height of the cylinder is equal to the circumference of the circular base. |
|
3stpes_13 | testmini | Properties of Cylinders | The lateral view of the cylinder is a rectangle ABCD as shown in the figure. Therefore, the front view of this cylinder is a (). | A. Cannot be determined; B. Circle; C. Rectangle; D. Square; E. No correct answer | D | 3steps_3 | 1,423 | Properties of Cylinders:
1. The top and bottom surfaces of a cylinder are called the bases.
2. A cylinder has a curved surface called the lateral surface.
3. The distance between the two bases of a cylinder is called the height.
4. A cylinder is formed by rotating a rectangle 180° around one of its edges.
5. The bases of a cylinder are circular.
6. The heights of all cylinders are equal. |
|
3stpes_13 | testmini | Area of Rectangles;Expanded View of Cylinders;Properties of Cylinders | The lateral view of the cylinder is a rectangle ABCD with an area of 12.56 cm² as shown in the figure. Therefore, the front view of this cylinder is a ().(π = 3.14) | A. Cannot be determined; B. Circle; C. Rectangle; D. Square; E. No correct answer | D | 3steps_multi | 1,588 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab.
Expanded View of Cylinders:
1. Formation of a Cylinder: A cylinder can be formed by rotating a rectangle around one of its edges, or by rolling a rectangle into a cylindrical shape.
2. Components of a Cylinder: A cylinder consists of bases and a lateral surface. The bases are two equal circular surfaces, referred to as the upper base and the lower base; the lateral surface is a curved surface that connects the upper and lower bases.
3. Unfolded Lateral Surface of a Cylinder: When the lateral surface of a cylinder is unfolded, it forms a rectangle or a square. If the unfolded surface is a rectangle, its length is the circumference of the circular base, and its width is the height of the cylinder; if the unfolded surface is a square, it means the height of the cylinder is equal to the circumference of the circular base.
Properties of Cylinders:
1. The top and bottom surfaces of a cylinder are called the bases.
2. A cylinder has a curved surface called the lateral surface.
3. The distance between the two bases of a cylinder is called the height.
4. A cylinder is formed by rotating a rectangle 180° around one of its edges.
5. The bases of a cylinder are circular.
6. The heights of all cylinders are equal. |
|
3stpes_14 | testmini | Perimeter of Squares | As shown in the figure, the perimeter of square ABCD is given. What is the length of its side in cm? | A. 4; B. 3; C. 2; D. 5; E. No correct answer | C | 3steps_1 | 1,094 | Perimeter of Squares:
1. A square has four equal sides, and the perimeter of a square = side length × 4 (C = 4a). |
|
3stpes_14 | testmini | Volume and Capacity of Rectangular Cuboids | As shown in the figure, the volume of the rectangular cuboid is 4 cm³, and its base is a square. What is the height of the rectangular cuboid in cm? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | D | 3steps_2 | 1,259 | Volume and Capacity of Rectangular Cuboids:
1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height.
2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh.
3. The volume of a rectangular cubiod is equal to the base area multiplied by the height.
4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods. |
|
3stpes_14 | testmini | Surface Area of Rectangular Cuboids | As shown in the figure, the rectangular cuboid has a square base. What is the surface area of the rectangular cuboid? ( ) cm² | A. 16; B. 12; C. 6; D. 4; E. No correct answer | A | 3steps_3 | 1,424 | Surface Area of Rectangular Cuboids:
1. Definition of Surface Area: The total area of the 6 faces of a rectangular cubiod or a cube is called its surface area.
2. Surface area of a rectangular cubiod = length × width × 2 + length × height × 2 + width × height × 2.
3. Surface area of a rectangular cubiod = (length × width + length × height + width × height) × 2.
4. Surface Area of a Rectangular Cubiod: If the letters a, b, and h represent the length, width, and height of a rectangular cubiod respectively, and S represents the surface area of the rectangular cubiod, then S = 2ab + 2ah + 2bh or S = 2(ab + ah + bh). |
|
3stpes_14 | testmini | Perimeter of Squares;Volume and Capacity of Rectangular Cuboids;Surface Area of Rectangular Cuboids | As shown in the figure, the volume of the rectangular cuboid is 4 cm³. Its base is a square. The surface area of the rectangular cuboid is () cm². | A. 16; B. 12; C. 6; D. 4; E. No correct answer | A | 3steps_multi | 1,589 | Perimeter of Squares:
1. A square has four equal sides, and the perimeter of a square = side length × 4 (C = 4a).
Volume and Capacity of Rectangular Cuboids:
1. The formula for calculating the volume of a rectangular cubiod: The volume of a rectangular cubed = length × width × height.
2. In terms of letters: If V represents the volume of the rectangular cubiod, and a, b, h represent the length, width, and height of the rectangular cubed respectively, then the formula for the volume of the rectangular cubed can be written as: V = abh.
3. The volume of a rectangular cubiod is equal to the base area multiplied by the height.
4. The capacity of a rectangular cubiod usually refers to the amount of space a rectangular container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
Surface Area of Rectangular Cuboids:
1. Definition of Surface Area: The total area of the 6 faces of a rectangular cubiod or a cube is called its surface area.
2. Surface area of a rectangular cubiod = length × width × 2 + length × height × 2 + width × height × 2.
3. Surface area of a rectangular cubiod = (length × width + length × height + width × height) × 2.
4. Surface Area of a Rectangular Cubiod: If the letters a, b, and h represent the length, width, and height of a rectangular cubiod respectively, and S represents the surface area of the rectangular cubiod, then S = 2ab + 2ah + 2bh or S = 2(ab + ah + bh). |
|
3stpes_15 | testmini | Properties and Understanding of Cubes | As shown in the figure, the rectangular cuboid can be divided into two identical cubes. The total length of the edges of each cube is marked in the figure. What is the edge length of one cube? ( ) cm | A. 4; B. 3; C. 2; D. 5; E. No correct answer | C | 3steps_1 | 1,095 | Properties and Understanding of Cubes:
1. A cube has six faces, all of which are squares.
2. All six faces of a cube have equal area.
3. A cube has 12 edges, and all edges are of equal length.
4. A cube has 8 vertices.
5. A cube can be considered a special type of rectangular cubiod. |
|
3stpes_15 | testmini | Combining and Dividing Solids | As shown in the figure, the rectangular cuboid can be split into two identical cubes. What are the length, width, and height of the original rectangular cuboid in cm? | A. 2, 2, 4; B. 2, 4, 2; C. 2, 2, 2; D. 4, 2, 2; E. No correct answer | D | 3steps_2 | 1,260 | Combining and Dividing Solids:
1. When two or more geometric solids are combined to form a new solid, the volume of the new solid equals the sum of the volumes of the original solids. However, the surface area of the new solid is less than the sum of the surface areas of the original solids. If the overlapping area is S, the decrease in surface area is 2S.
2. When a geometric solid is cut into parts, the sum of the volumes of the parts equals the volume of the original solid. However, the sum of the surface areas of the parts is greater than the surface area of the original solid. If the cutting area is S, the increase in surface area is 2S. |
|
3stpes_15 | testmini | Surface Area of Rectangular Cuboids | As shown in the figure, the rectangular cuboid can be split into two identical cubes. What is the surface area of the rectangular cuboid? ( ) cm² | A. 48; B. 40; C. 32; D. 20; E. No correct answer | B | 3steps_3 | 1,425 | Surface Area of Rectangular Cuboids:
1. Definition of Surface Area: The total area of the 6 faces of a rectangular cubiod or a cube is called its surface area.
2. Surface area of a rectangular cubiod = length × width × 2 + length × height × 2 + width × height × 2.
3. Surface area of a rectangular cubiod = (length × width + length × height + width × height) × 2.
4. Surface Area of a Rectangular Cubiod: If the letters a, b, and h represent the length, width, and height of a rectangular cubiod respectively, and S represents the surface area of the rectangular cubiod, then S = 2ab + 2ah + 2bh or S = 2(ab + ah + bh). |
|
3stpes_15 | testmini | Properties and Understanding of Cubes;Combining and Dividing Solids;Surface Area of Rectangular Cuboids | As shown in the figure, the rectangular cuboid can be split into two identical cubes. The total length of the edges of each cube is marked in the figure. What is the surface area of this rectangular cuboid? ( ) cm² | A. 48; B. 40; C. 32; D. 20; E. No correct answer | B | 3steps_multi | 1,590 | Properties and Understanding of Cubes:
1. A cube has six faces, all of which are squares.
2. All six faces of a cube have equal area.
3. A cube has 12 edges, and all edges are of equal length.
4. A cube has 8 vertices.
5. A cube can be considered a special type of rectangular cubiod.
Combining and Dividing Solids:
1. When two or more geometric solids are combined to form a new solid, the volume of the new solid equals the sum of the volumes of the original solids. However, the surface area of the new solid is less than the sum of the surface areas of the original solids. If the overlapping area is S, the decrease in surface area is 2S.
2. When a geometric solid is cut into parts, the sum of the volumes of the parts equals the volume of the original solid. However, the sum of the surface areas of the parts is greater than the surface area of the original solid. If the cutting area is S, the increase in surface area is 2S.
Surface Area of Rectangular Cuboids:
1. Definition of Surface Area: The total area of the 6 faces of a rectangular cubiod or a cube is called its surface area.
2. Surface area of a rectangular cubiod = length × width × 2 + length × height × 2 + width × height × 2.
3. Surface area of a rectangular cubiod = (length × width + length × height + width × height) × 2.
4. Surface Area of a Rectangular Cubiod: If the letters a, b, and h represent the length, width, and height of a rectangular cubiod respectively, and S represents the surface area of the rectangular cubiod, then S = 2ab + 2ah + 2bh or S = 2(ab + ah + bh). |
|
3stpes_16 | testmini | Volume and Capacity of Cones | As shown in the figure, the volume and height of the cone are given. What is the radius of the base of this cone in cm?(π = 3.14) | A. 4; B. 3; C. 2; D. 1; E. No correct answer | D | 3steps_1 | 1,096 | Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_16 | testmini | Properties of Cones | As shown in the figure, the height and the base radius of the cone are given. Then the front view of the cone is a ( ) | A. Square; B. Rectangle; C. Equilateral triangle; D. Isosceles triangle; E. No correct answer | D | 3steps_2 | 1,261 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
|
3stpes_16 | testmini | Area of Triangles | As shown in the figure, the front view of the cone is the isosceles triangle below. The area of this triangle is () cm². | A. 4; B. 3; C. 2; D. 1; E. No correct answer | B | 3steps_3 | 1,426 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_16 | testmini | Volume and Capacity of Cones;Properties of Cones;Area of Triangles | As shown in the figure, the volume and height of the cone are given. What is the area of the front view of this cone? ( ) cm².(π = 3.14) | A. 4; B. 3; C. 2; D. 1; E. No correct answer | B | 3steps_multi | 1,591 | Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14.
Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length.
Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_17 | testmini | Area of Triangles | As shown in the figure, the area of triangle ABC is 48 cm², and it is the front view of a cone. What is the length of BC in cm? | A. 16; B. 8; C. 6; D. 4; E. No correct answer | B | 3steps_1 | 1,097 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_17 | testmini | Properties of Cones | As shown in the figure, triangle ABC is the front view of a cone. What is the height of this cone in cm, and what is the radius of its base in cm? | A. 12, 4; B. 12, 2; C. 12, 8; D. 10, 4; E. No correct answer | A | 3steps_2 | 1,262 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
|
3stpes_17 | testmini | Volume and Capacity of Cones | The height and the radius of the base of a cone are shown in the diagram. What is the volume of this cone in cm³? | A. 24π; B. 16π; C. 32π; D. 64π; E. No correct answer | D | 3steps_3 | 1,427 | Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_17 | testmini | Area of Triangles;Properties of Cones;Volume and Capacity of Cones | As shown in the figure, triangle ABC with an area of 48 cm² is the front view of a cone. What is the volume of this cone in cm³? | A. 24π; B. 16π; C. 32π; D. 64π; E. No correct answer | D | 3steps_multi | 1,592 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined.
Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length.
Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_18 | testmini | Area of Circles | As shown in the figure, the central angle of the sector is 90°, and its area is equal to the area of the circle shown below. What is the area of the sector in cm²?(Use π=3.14) | A. 50.24; B. 3.14; C. 12.56; D. 6.28; E. No correct answer | B | 3steps_1 | 1,098 | Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14. |
|
3stpes_18 | testmini | Area of Sectors | As shown in the figure, the area of the sector is marked. What is the radius of this sector in cm? (Use π=3.14) | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_2 | 1,263 | Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14. |
|
3stpes_18 | testmini | Area of Triangles | The radius of the sector is shown in the figure, and the central angle is 90°. What is the area of triangle ABC? ( ) cm² | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_3 | 1,428 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_18 | testmini | Area of Circles;Area of Sectors;Area of Triangles | Given that the area of the sector with a central angle of 90° is equal to the area of a circle on the left, connect points A and C. What is the area of triangle ABC in cm²?（Use π=3.14） | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_multi | 1,593 | Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14.
Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14.
Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_19 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, ABCD is a parallelogram. What is the measure of ∠B? ( )° | A. 45; B. 60; C. 30; D. 90; E. No correct answer | C | 3steps_1 | 1,099 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral. |
|
3stpes_19 | testmini | Area of Sectors | As shown in the figure, a sector is drawn with B as the vertex and AB as the radius, just passing through point C. Given that the area of this sector is 37.68 cm², what is the length of AB=BC= ( ) cm?(Use π=3.14) | A. 12; B. 8; C. 6; D. 4; E. No correct answer | A | 3steps_2 | 1,264 | Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14. |
|
3stpes_19 | testmini | Perimeter of Parallelograms | As shown in the figure, quadrilateral ABCD is a parallelogram. What is the perimeter of this parallelogram in cm? | A. 48; B. 40; C. 32; D. 20; E. No correct answer | A | 3steps_3 | 1,429 | Perimeter of Parallelograms:
1. A parallelogram has equal opposite sides, and its perimeter is twice the sum of its adjacent sides. The formula for the perimeter is C = 2(a + b), where a and b are the lengths of the sides of the parallelogram. |
|
3stpes_19 | testmini | Properties and Understanding of Parallelograms;Area of Sectors;Perimeter of Parallelograms | As shown in the figure, ABCD is a parallelogram. Using B as the vertex and AB as the radius, a sector is drawn that just passes through point C. Given that the area of this sector is 37.68 cm², what is the perimeter of this parallelogram in cm?(Use π=3.14) | A. 48; B. 40; C. 32; D. 20; E. No correct answer | A | 3steps_multi | 1,594 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral.
Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14.
Perimeter of Parallelograms:
1. A parallelogram has equal opposite sides, and its perimeter is twice the sum of its adjacent sides. The formula for the perimeter is C = 2(a + b), where a and b are the lengths of the sides of the parallelogram. |
|
3stpes_20 | testmini | Properties and Understanding of Trapezoids | As shown in the figure, quadrilateral ABCD is an isosceles trapezoid. What is the length relationship between AB and CD? ( ) | A. AB = CD; B. AB > CD; C. AB < CD; D. No correct answer | A | 3steps_1 | 1,100 | Properties and Understanding of Trapezoids:
1. A trapezoid (or trapezium) is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called the bases of the trapezoid: the longer base is called the lower base, and the shorter base is called the upper base; the other two sides are called the legs; the perpendicular segment between the two bases is called the height of the trapezoid.
2. A trapezoid with one leg perpendicular to the bases is called a right trapezoid.
3. A trapezoid with both legs equal in length is called an isosceles trapezoid.
4. The height of a trapezoid is the distance between the upper base and the lower base. |
|
3stpes_20 | testmini | Folding Problems of Figures | As shown in the figure, point E is taken on BC, and quadrilateral ABCD is folded along DE so that the folded point C coincides exactly with point A. What is the relationship between the lengths of CD and AD? | A. AD = CD; B. AD > CD; C. AD < CD; D. No correct answer | A | 3steps_2 | 1,265 | Folding Problems of Figures:
1. The folded figure exhibits the properties of a symmetrical figure, with the fold line as the axis of symmetry, and the two parts of the figure before and after folding are symmetrical about the fold line. |
|
3stpes_20 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, DC = AD = AB, and quadrilateral ABED is a parallelogram. What is the relationship between the lengths of DE and AD? | A. DE=AD; B. DE>AD; C. DE<AD; D. No correct answer | A | 3steps_3 | 1,430 | Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral. |
|
3stpes_20 | testmini | Properties and Understanding of Trapezoids;Folding Problems of Figures;Properties and Understanding of Parallelograms | As shown in the figure, quadrilateral ABCD is an isosceles trapezoid. Point E is taken on BC, and the figure is folded along DE such that point C coincides exactly with point A after folding. At this time, a parallelogram ABED is formed. The relationship between the lengths of DE and AD is ( ) | A. DE=AD; B. DE>AD; C. DE<AD; D. No correct answer | A | 3steps_multi | 1,595 | Properties and Understanding of Trapezoids:
1. A trapezoid (or trapezium) is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called the bases of the trapezoid: the longer base is called the lower base, and the shorter base is called the upper base; the other two sides are called the legs; the perpendicular segment between the two bases is called the height of the trapezoid.
2. A trapezoid with one leg perpendicular to the bases is called a right trapezoid.
3. A trapezoid with both legs equal in length is called an isosceles trapezoid.
4. The height of a trapezoid is the distance between the upper base and the lower base.
Folding Problems of Figures:
1. The folded figure exhibits the properties of a symmetrical figure, with the fold line as the axis of symmetry, and the two parts of the figure before and after folding are symmetrical about the fold line.
Properties and Understanding of Parallelograms:
1. A parallelogram is a closed shape in a plane formed by two pairs of parallel line segments.
2. A quadrilateral with two pairs of opposite sides parallel is called a parallelogram.
3. Rectangles, rhombuses, and squares are special types of parallelograms.
4. In a parallelogram, both pairs of opposite sides are equal in length, both pairs of opposite angles are equal, adjacent angles are supplementary, the height (distance between the parallel lines) is the same everywhere, and the diagonals bisect each other.
5. The height of a parallelogram is the perpendicular segment drawn from a point on one side to the opposite side, and this segment is called the height of the parallelogram.
6. To draw a parallelogram, first draw two parallel and equal-length line segments, then connect the endpoints of these two segments to form a quadrilateral. |
|
3stpes_21 | testmini | Expanded View of Cylinders | A rectangular steel plate with a length of 20.56 cm is shown in the diagram. The shaded part obtained by cutting the plate can be used to make a cylinder. What is the radius of the circular shaded part in cm?(Use π = 3.14) | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_1 | 1,101 | Expanded View of Cylinders:
1. Formation of a Cylinder: A cylinder can be formed by rotating a rectangle around one of its edges, or by rolling a rectangle into a cylindrical shape.
2. Components of a Cylinder: A cylinder consists of bases and a lateral surface. The bases are two equal circular surfaces, referred to as the upper base and the lower base; the lateral surface is a curved surface that connects the upper and lower bases.
3. Unfolded Lateral Surface of a Cylinder: When the lateral surface of a cylinder is unfolded, it forms a rectangle or a square. If the unfolded surface is a rectangle, its length is the circumference of the circular base, and its width is the height of the cylinder; if the unfolded surface is a square, it means the height of the cylinder is equal to the circumference of the circular base. |
|
3stpes_21 | testmini | Area of Circles | As shown in the figure, the area of the shaded part of the circle is () cm² .(Use π=3.14) | A. 50.24; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | C | 3steps_2 | 1,266 | Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14. |
|
3stpes_21 | testmini | Volume and Capacity of Cylinders | A rectangular steel plate is shown in the figure. The shaded part obtained by cutting can be used to make a cylinder. The area of the circular shaded part is 12.56 cm². What is the volume of this cylinder in cm³?(Use π = 3.14) | A. 50.24; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | A | 3steps_3 | 1,431 | Volume and Capacity of Cylinders:
1. The formula for calculating the volume of a cylinder: The volume of a cylinder = base area × height.
2. In terms of letters: If V represents the volume of the cylinder, where r is the radius of the base and h is the height of the cylinder, then the formula for the volume of the cylinder is: V = πr²h.
3. The capacity of a cylinder usually refers to the amount of space a cylindrical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
4. π is generally taken as 3.14. |
|
3stpes_21 | testmini | Expanded View of Cylinders;Area of Circles;Volume and Capacity of Cylinders | A rectangular steel plate with a length of 20.56 cm is shown in the diagram. The shaded portion obtained after cutting can be used to make a cylinder. What is the volume of this cylinder in cm³? (Use π = 3.14) | A. 50.24; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | A | 3steps_multi | 1,596 | Expanded View of Cylinders:
1. Formation of a Cylinder: A cylinder can be formed by rotating a rectangle around one of its edges, or by rolling a rectangle into a cylindrical shape.
2. Components of a Cylinder: A cylinder consists of bases and a lateral surface. The bases are two equal circular surfaces, referred to as the upper base and the lower base; the lateral surface is a curved surface that connects the upper and lower bases.
3. Unfolded Lateral Surface of a Cylinder: When the lateral surface of a cylinder is unfolded, it forms a rectangle or a square. If the unfolded surface is a rectangle, its length is the circumference of the circular base, and its width is the height of the cylinder; if the unfolded surface is a square, it means the height of the cylinder is equal to the circumference of the circular base.
Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14.
Volume and Capacity of Cylinders:
1. The formula for calculating the volume of a cylinder: The volume of a cylinder = base area × height.
2. In terms of letters: If V represents the volume of the cylinder, where r is the radius of the base and h is the height of the cylinder, then the formula for the volume of the cylinder is: V = πr²h.
3. The capacity of a cylinder usually refers to the amount of space a cylindrical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
4. π is generally taken as 3.14. |
|
3stpes_22 | testmini | Area of Triangles | The area of triangle ABC is 12 cm². What is the length of BC in cm? | A. 4; B. 5; C. 6; D. 3; E. No correct answer | C | 3steps_1 | 1,102 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined. |
|
3stpes_22 | testmini | Area of Circles | As shown in the figure, what is the area of the circle with diameter BC in cm²? (Use π = 3.14) ( ) | A. 28.26; B. 25.12; C. 12.56; D. 6.28; E. No correct answer | A | 3steps_2 | 1,267 | Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14. |
|
3stpes_22 | testmini | Volume and Capacity of Cones | As shown in the figure, triangle ABC is the front view of a cone. The base area of the cone is 28.26 cm². What is the volume of the cone ( ) cm³? | A. 50.24; B. 37.68; C. 125.6; D. 62.8; E. No correct answer | B | 3steps_3 | 1,432 | Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_22 | testmini | Area of Triangles;Area of Circles;Volume and Capacity of Cones | As shown in the figure, triangle ABC is the front view of a cone. The area of triangle ABC is 12 cm². What is the volume of the cone with triangle ABC as its front view? ( ) cm³?(π = 3.14) | A. 50.24; B. 37.68; C. 125.6; D. 62.8; E. No correct answer | B | 3steps_multi | 1,597 | Area of Triangles:
1. If the base is a and the height is h, then the area of a triangle is S = 1/2 ah.
2. The area of a right triangle is equal to the product of its two legs divided by two.
3. If the area and height of a triangle are known, its base length can be calculated. Similarly, if the area and base length of a triangle are known, its height can be determined.
Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14.
Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
|
3stpes_23 | testmini | Circumference of Circles | As shown in the figure, two identical largest circles are cut out from a rectangular piece of paper. The circumference of each circle is 12.56 cm. What is the radius of each circle in cm? (Use π = 3.14) | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_1 | 1,103 | Circumference of Circles:
1. A line segment passing through the center of a circle with both endpoints on the circle is called the diameter, usually denoted by the letter d. A line segment connecting the center of the circle to any point on the circle is called the radius, usually denoted by the letter r.
2. The circumference of a circle = diameter × pi (C = πd) or the circumference of a circle = 2 × radius × pi (C = 2πr).
3. The circumference of a semicircle can be calculated using the formula C = πr + 2r.
4. In the same circle or congruent circles, there are an infinite number of radii and an infinite number of diameters. All radii and diameters in the same circle are equal. The length of the diameter is twice the length of the radius, and the length of the radius is half the length of the diameter. |
|
3stpes_23 | testmini | Understanding Circles | As shown in the figure, two identical largest circles can be cut out from a rectangular piece of paper. What are the side lengths of the rectangle, AB = ( ) cm, AD = ( ) cm? | A. 2, 4; B. 8, 4; C. 4, 4; D. 4, 8; E. No correct answer | D | 3steps_2 | 1,268 | Understanding Circles:
1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point.
2. A circle is an axisymmetric figure.
3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r.
4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius.
5. All radii of a circle are equal in length, and all diameters are equal in length.
6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle. |
|
3stpes_23 | testmini | Perimeter of Rectangles | As shown in the figure, two identical largest circles can be cut out from the rectangular paper. What is the perimeter of the rectangle ABCD? (Use π = 3.14) ( ) cm | A. 48; B. 40; C. 32; D. 24; E. No correct answer | D | 3steps_3 | 1,433 | Perimeter of Rectangles:
1. A rectangle has equal opposite sides, and the perimeter of a rectangle = (length + width) × 2 (C = 2(a+b)). |
|
3stpes_23 | testmini | Circumference of Circles;Understanding Circles;Perimeter of Rectangles | As shown in the figure, two identical largest circles are cut out from a rectangular piece of paper. The circumference of one of the circles is 12.56 cm. What is the perimeter of the rectangle? (Use π = 3.14) ( ) cm² | A. 48; B. 40; C. 32; D. 24; E. No correct answer | D | 3steps_multi | 1,598 | Circumference of Circles:
1. A line segment passing through the center of a circle with both endpoints on the circle is called the diameter, usually denoted by the letter d. A line segment connecting the center of the circle to any point on the circle is called the radius, usually denoted by the letter r.
2. The circumference of a circle = diameter × pi (C = πd) or the circumference of a circle = 2 × radius × pi (C = 2πr).
3. The circumference of a semicircle can be calculated using the formula C = πr + 2r.
4. In the same circle or congruent circles, there are an infinite number of radii and an infinite number of diameters. All radii and diameters in the same circle are equal. The length of the diameter is twice the length of the radius, and the length of the radius is half the length of the diameter.
Understanding Circles:
1. In a plane, a circle is defined as the set of all points that are at a fixed distance from a fixed point.
2. A circle is an axisymmetric figure.
3. The center of the circle is called the center, usually denoted by the letter O. A line segment connecting the center to any point on the circle is called the radius, usually denoted by the letter r.
4. A line segment that passes through the center and has its endpoints on the circle is called the diameter, usually denoted by the letter d. The diameter is the longest line segment within the circle, and it is twice the length of the radius.
5. All radii of a circle are equal in length, and all diameters are equal in length.
6. Method of drawing a circle: Use a compass. The distance between the legs of the compass represents the radius. After setting the center and the radius, rotate the compass to draw a circle.
Perimeter of Rectangles:
1. A rectangle has equal opposite sides, and the perimeter of a rectangle = (length + width) × 2 (C = 2(a+b)). |
|
3stpes_24 | testmini | Area of Squares | As shown in the figure, the area of square ABCD is () cm². | A. 16; B. 8; C. 6; D. 4; E. No correct answer | A | 3steps_1 | 1,104 | Area of Squares:
1. The area of a square is equal to the square of its side length: S = a * a.
2. The area of a square is equal to the square of the length of its diagonal divided by two. |
|
3stpes_24 | testmini | Area of Circles | As shown in the figure, the area of square ABCD is 16 cm². A circle with a radius of 2 cm is drawn inside the square. What is the area of the shaded portion in the lower right corner? (Use π = 3.14) | A. 0.86; B. 5.44; C. 12.56; D. Cannot be determined; E. No correct answer | A | 3steps_2 | 1,269 | Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14. |
|
3stpes_24 | testmini | Area of Sectors | As shown in the figure, the area of square ABCD is 16 cm². A circle with a radius of 2 cm is drawn inside the square. The area of the shaded region S1 in the figure is 0.86 cm². Then, using point A as the center and AB as the radius, a sector ABD is drawn, forming another irregular shaded region S2. What is the area of the shaded region S2? (Use π = 3.14) | A. 50.24; B. 3.44; C. 12.56; D. 2.58; E. No correct answer | D | 3steps_3 | 1,434 | Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14. |
|
3stpes_24 | testmini | Area of Squares;Area of Circles;Area of Sectors | As shown in the figure, quadrilateral ABCD is a square. Inside the square, a circle with a radius of 2 cm is drawn, forming the shaded area S1. Then, using point A as the center and AB as the radius, a sector ABD is drawn, forming an irregular shaded area S2. The area of the shaded part S2 is () cm².(Use π = 3.14) | A. 50.24; B. 3.44; C. 12.56; D. 2.58; E. No correct answer | D | 3steps_multi | 1,599 | Area of Squares:
1. The area of a square is equal to the square of its side length: S = a * a.
2. The area of a square is equal to the square of the length of its diagonal divided by two.
Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14.
Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14. |
|
3stpes_25 | testmini | Understanding Triangular Rulers | Placing a triangular ruler in the position shown in the figure, the two edges of the triangular ruler exactly coincide with the two edges of ∠A. What is the measure of ∠A? ( )° | A. 45; B. 60; C. 30; D. 90; E. No correct answer | C | 3steps_1 | 1,105 | Understanding Triangular Rulers:
1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions.
2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°. |
|
3stpes_25 | testmini | Properties and Understanding of Triangles | As shown in the figure, triangle ABC is an isosceles triangle. A triangular ruler is placed at ∠A, and the two edges of the triangular ruler coincide exactly with the two sides of ∠A. What is ∠C equal to? ( )° | A. 45; B. 60; C. 30; D. 90; E. No correct answer | C | 3steps_2 | 1,270 | Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change. |
|
3stpes_25 | testmini | Sum of Interior Angles of Triangles | As shown in the figure, triangle ABC is an isosceles triangle, ∠B = ( )° | A. 30; B. 60; C. 120; D. 150; E. No correct answer | C | 3steps_3 | 1,435 | Sum of Interior Angles of Triangles:
1. The sum of the interior angles of any triangle is 180°.
2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. |
|
3stpes_25 | testmini | Understanding Triangular Rulers;Properties and Understanding of Triangles;Sum of Interior Angles of Triangles | As shown in the figure, triangle ABC is an isosceles triangle, and the two sides of the triangular ruler coincide exactly with the two sides of ∠A. What is the measure of ∠B? ( )° | A. 30; B. 60; C. 120; D. 150; E. No correct answer | C | 3steps_multi | 1,600 | Understanding Triangular Rulers:
1. A triangular ruler is a triangular-shaped tool commonly used for measuring angles, drawing straight lines, and performing geometric constructions.
2. There are two types of triangular ruler: Isosceles right trianglular ruler: One angle is 90°, and the other two angles are 45° and 45°. Scalene right triangular ruler: One angle is 90°, and the other two angles are 30° and 60°.
Properties and Understanding of Triangles:
1. A triangle is a closed geometric figure formed by three line segments connected end-to-end in sequence.
2. By angle classification: Acute triangle: All three interior angles of the triangle are less than 90 degrees. Right triangle: One of the three interior angles of the triangle is exactly 90 degrees. Obtuse triangle: One of the three interior angles of the triangle is greater than 90 degrees.
3. By side classification: Scalene triangle: A triangle where all three sides are of different lengths. Isosceles triangle: A triangle with two equal sides. Equilateral triangle: A triangle with all three sides equal.
4. The sum of the interior angles of a triangle in a plane is 180 degrees.
5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side, and the difference between the lengths of any two sides is less than the length of the third side.
6. The height of a triangle is the perpendicular line segment drawn from a vertex to the opposite side.
7. To draw the height of a triangle: Choose one side as the base. Find the vertex opposite the base. Using a ruler and compass, draw a perpendicular line from the vertex to the base. Ensure to mark the right angle symbol, indicating it is the height.
8. Once the lengths of the three sides of a triangle are determined, the triangle is fixed. The angles and the area enclosed by the three sides do not change.
Sum of Interior Angles of Triangles:
1. The sum of the interior angles of any triangle is 180°.
2. According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. |
End of preview. Expand
in Dataset Viewer.
README.md exists but content is empty.
Use the Edit dataset card button to edit it.
- Downloads last month
- 36