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3stpes_26 | testmini | Properties and Understanding of Trapezoids | In the quadrilateral ABCD shown in the figure, ABCD is a right trapezoid, and line segment BE is perpendicular to CD. What is the length of BE in cm? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_1 | 1,106 | 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. |
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3stpes_26 | testmini | Properties and Understanding of Triangles | In triangle BEC, line segment BE is perpendicular to CD, then CE = ( ) cm | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_2 | 1,271 | 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. |
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3stpes_26 | testmini | Properties and Understanding of Rectangles | As shown in the figure, quadrilateral ABDE is a rectangle. What is the length of CD in cm? | A. 5; B. 4; C. 3; D. 2; E. No correct answer | A | 3steps_3 | 1,436 | Properties and Understanding of Rectangles:
1. A rectangle is a plane figure and a parallelogram with one right angle. A rectangle is also defined as a parallelogram with all four angles being right angles. A square is a special type of rectangle where all four sides are equal in length.
2. Properties of a rectangle: The two diagonals are equal in length; the two diagonals bisect each other; both pairs of opposite sides are parallel; both pairs of opposite sides are equal in length; all four angles are right angles; there are two axes of symmetry; it is unstable (prone to deformation); the square of the length of a rectangle's diagonal is equal to the sum of the squares of its two sides; the quadrilateral formed by sequentially connecting the midpoints of a rectangle's sides is a rhombus. |
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3stpes_26 | testmini | Properties and Understanding of Trapezoids;Properties and Understanding of Triangles;Properties and Understanding of Rectangles | As shown in the figure, ABCD is a right trapezoid with a rectangle ABDE inside it. What is the length of CD? ( ) cm | A. 5; B. 4; C. 3; D. 2; E. No correct answer | A | 3steps_multi | 1,601 | 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.
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.
Properties and Understanding of Rectangles:
1. A rectangle is a plane figure and a parallelogram with one right angle. A rectangle is also defined as a parallelogram with all four angles being right angles. A square is a special type of rectangle where all four sides are equal in length.
2. Properties of a rectangle: The two diagonals are equal in length; the two diagonals bisect each other; both pairs of opposite sides are parallel; both pairs of opposite sides are equal in length; all four angles are right angles; there are two axes of symmetry; it is unstable (prone to deformation); the square of the length of a rectangle's diagonal is equal to the sum of the squares of its two sides; the quadrilateral formed by sequentially connecting the midpoints of a rectangle's sides is a rhombus. |
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3stpes_27 | testmini | Properties and Understanding of Rectangles | In rectangle ABCD, the length of BC is () cm. | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_1 | 1,107 | Properties and Understanding of Rectangles:
1. A rectangle is a plane figure and a parallelogram with one right angle. A rectangle is also defined as a parallelogram with all four angles being right angles. A square is a special type of rectangle where all four sides are equal in length.
2. Properties of a rectangle: The two diagonals are equal in length; the two diagonals bisect each other; both pairs of opposite sides are parallel; both pairs of opposite sides are equal in length; all four angles are right angles; there are two axes of symmetry; it is unstable (prone to deformation); the square of the length of a rectangle's diagonal is equal to the sum of the squares of its two sides; the quadrilateral formed by sequentially connecting the midpoints of a rectangle's sides is a rhombus. |
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3stpes_27 | testmini | Understanding Sectors | In rectangle ABCD, DAE and CBE are two sectors with radii AD and BC respectively. What is the length of AB? () | A. 4; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_2 | 1,272 | 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". |
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3stpes_27 | testmini | Perimeter of Rectangles | As shown in the figure, DAE and CBE are two sectors with radii AD and BC, respectively. What is the perimeter of rectangle ABCD? ( ) cm | A. 16; B. 12; C. 6; D. 4; E. No correct answer | B | 3steps_3 | 1,437 | Perimeter of Rectangles:
1. A rectangle has equal opposite sides, and the perimeter of a rectangle = (length + width) × 2 (C = 2(a+b)). |
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3stpes_27 | testmini | Properties and Understanding of Rectangles;Understanding Sectors;Perimeter of Rectangles | In rectangle ABCD, DAE and CBE are two sectors with radii AD and BC, respectively. What is the perimeter of rectangle ABCD? ( ) cm | A. 16; B. 12; C. 6; D. 4; E. No correct answer | B | 3steps_multi | 1,602 | Properties and Understanding of Rectangles:
1. A rectangle is a plane figure and a parallelogram with one right angle. A rectangle is also defined as a parallelogram with all four angles being right angles. A square is a special type of rectangle where all four sides are equal in length.
2. Properties of a rectangle: The two diagonals are equal in length; the two diagonals bisect each other; both pairs of opposite sides are parallel; both pairs of opposite sides are equal in length; all four angles are right angles; there are two axes of symmetry; it is unstable (prone to deformation); the square of the length of a rectangle's diagonal is equal to the sum of the squares of its two sides; the quadrilateral formed by sequentially connecting the midpoints of a rectangle's sides is a rhombus.
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".
Perimeter of Rectangles:
1. A rectangle has equal opposite sides, and the perimeter of a rectangle = (length + width) × 2 (C = 2(a+b)). |
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3stpes_28 | testmini | Properties and Understanding of Cubes | If the height of a rectangular cuboid is increased by 4 cm, it becomes a cube. What is the shape of the base of this rectangular cuboid? ( ) | A. Square; B. Rectangle; C. Parallelogram; D. Cannot be determined; E. No correct answer | A | 3steps_1 | 1,108 | 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. |
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3stpes_28 | testmini | Properties and Understanding of Rectangular Cuboids | The base of a rectangular cuboid is a square. If the height of the rectangular cuboid is increased by 4 cm, and its surface area also increases by 128 cm², what is the side length of the base of the rectangular cuboid in cm? | A. 12; B. 8; C. 6; D. 4; E. No correct answer | B | 3steps_2 | 1,273 | Properties and Understanding of Rectangular Cuboids:
1. A rectangular cubiod has six faces, all of which are rectangles.
2. The opposite faces of a rectangular cubed have equal area, and of the 12 edges, the lengths of the 4 opposite edges are equal.
3. A rectangular cubiod has 8 vertices.
4. The lengths of the three edges intersecting at one vertex of a rectangular cubiod are called length, width, and height.
5. The edge where two faces of a rectangular cubiod meet is called an edge.
6. The point where three edges of a rectangular cubiod meet is called a vertex.
7. When placed on a table, a rectangular cubiod can show at most three faces.
8. The total area of the six faces of a rectangular cubiod is called its surface area. |
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3stpes_28 | testmini | Surface Area of Cubes | A rectangular cuboid with a square base, if its height is increased by 4 cm, it becomes a cube. What is the original surface area of the rectangular cuboid in cm²? | A. 512; B. 64; C. 128; D. 256; E. No correct answer | D | 3steps_3 | 1,438 | Surface Area of Cubes:
1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s.
2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6.
3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. |
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3stpes_28 | testmini | Properties and Understanding of Cubes;Properties and Understanding of Rectangular Cuboids;Surface Area of Cubes | A rectangular cuboid, if its height is increased by 4 cm, becomes a cube. At this point, the surface area is increased by 128 cm² compared to the original. What was the original surface area of the rectangular cuboid in square centimeters? | A. 512; B. 64; C. 128; D. 256; E. No correct answer | D | 3steps_multi | 1,603 | 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.
Properties and Understanding of Rectangular Cuboids:
1. A rectangular cubiod has six faces, all of which are rectangles.
2. The opposite faces of a rectangular cubed have equal area, and of the 12 edges, the lengths of the 4 opposite edges are equal.
3. A rectangular cubiod has 8 vertices.
4. The lengths of the three edges intersecting at one vertex of a rectangular cubiod are called length, width, and height.
5. The edge where two faces of a rectangular cubiod meet is called an edge.
6. The point where three edges of a rectangular cubiod meet is called a vertex.
7. When placed on a table, a rectangular cubiod can show at most three faces.
8. The total area of the six faces of a rectangular cubiod is called its surface area.
Surface Area of Cubes:
1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s.
2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6.
3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. |
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3stpes_29 | testmini | Circumference of Circles | The circumference of the circle in the diagram is () cm.(Use π = 3.14) | A. 50.24; B. 3.14; C. 12.56; D. 6.28; E. No correct answer | C | 3steps_1 | 1,109 | 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. |
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3stpes_29 | testmini | Expanded View of Cylinders | As shown in the figure, the diagram is the unfolded view of a cylinder. The circumference of each of the two circles in the diagram is 12.56 cm. What is the length of the rectangle in the diagram? ( ) cm | A. 50.24; B. 3.14; C. 12.56; D. 6.28; E. No correct answer | C | 3steps_2 | 1,274 | 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. |
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3stpes_29 | testmini | Area of Rectangles | As shown in the figure, the length of the rectangle is 12.56 cm. What is the area of the rectangle in cm²?(Use π = 3.14) | A. 50.24; B. 25.12; C. 12.56; D. 75.36; E. No correct answer | D | 3steps_3 | 1,439 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
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3stpes_29 | testmini | Circumference of Circles;Expanded View of Cylinders;Area of Rectangles | As shown in the figure, the diagram is the unfolded view of a cylinder. What is the lateral surface area of this cylinder?(Use π = 3.14) ( ) cm² | A. 50.24; B. 25.12; C. 12.56; D. 75.36; E. No correct answer | D | 3steps_multi | 1,604 | 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.
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 Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
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3stpes_30 | testmini | Expanded View of Cylinders | If the height of a cylinder is reduced by 4 centimeters, and the surface area is reduced by 50.24 cm², what is the circumference of the base of the cylinder in centimeters? | A. 50.24; B. 3.14; C. 12.56; D. 6.28; E. No correct answer | C | 3steps_1 | 1,110 | 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. |
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3stpes_30 | testmini | Circumference of Circles | As shown in the figure, the perimeter of the base of the solid is 12.56 cm. What is the radius of the base in cm? (Use π = 3.14) | A. 4; B. 3; C. 2; D. 1; E. No correct answer | C | 3steps_2 | 1,275 | 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. |
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3stpes_30 | testmini | Surface Area of Cylinders | As shown in the figure, the circumference of the base of a cylinder is equal to its height. The circumference of the base is 12.56 cm. What is the surface area of the cylinder? (Use π = 3.14) | A. 502.4; B. 251.2; C. 125.6; D. 182.8; E. No correct answer | D | 3steps_3 | 1,440 | 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. |
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3stpes_30 | testmini | Expanded View of Cylinders;Circumference of Circles;Surface Area of Cylinders | As shown in the figure, the circumference of the base of a cylinder is equal to its height. If the height is reduced by 4 centimeters, the surface area decreases by 50.24 square centimeters. What was the original surface area of this cylinder? | A. 502.4; B. 251.2; C. 125.6; D. 182.8; E. No correct answer | D | 3steps_multi | 1,605 | 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.
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.
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. |
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3stpes_31 | testmini | Area of Circles | As shown in the figure, there is a circle enclosed by a wire, and its area is marked in the figure. What is the radius of the circle? (Unit: cm) | A. 5; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_1 | 1,111 | 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. |
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3stpes_31 | testmini | Circumference of Circles | As shown in the figure, there is a circle enclosed by a wire, with its radius marked in the diagram. What is the circumference of this circle? | A. 10π; B. 3π; C. 2π; D. π; E. No correct answer | A | 3steps_2 | 1,276 | 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. |
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3stpes_31 | testmini | Perimeter of Trapezoids | As shown in the figure, a wire is used to form the circle on the left. The same wire, with a length of 10π, can also be used to form the isosceles trapezoid on the right. What is the length of the legs of this trapezoid in π? (Unit: cm) | A. 4; B. 5; C. 3; D. No correct answer | C | 3steps_3 | 1,441 | 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. |
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3stpes_31 | testmini | Area of Circles;Circumference of Circles;Perimeter of Trapezoids | As shown in the figure, there is a circle enclosed by a wire, and its area is as shown. Now, this wire is reformed into an isosceles trapezoid with the top and bottom as shown in the figure. What is the length of the leg of this trapezoid in π? (Unit: cm) | A. 4; B. 5; C. 3; D. No correct answer | C | 3steps_multi | 1,606 | 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.
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.
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. |
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3stpes_32 | testmini | Understanding Triangular Rulers | As shown in the figure, a triangular ruler is stacked with three circular paper pieces. What is the relationship between the lengths of the two perpendicular sides of the triangular ruler? | A. Equal; B. Not equal; C. Cannot be determined; D. No correct answer | A | 3steps_1 | 1,112 | 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°. |
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3stpes_32 | testmini | Circumference of Circles | As shown in the figure, a right-angled triangle is overlapped with three circular paper pieces. The lengths of the two legs of the triangular ruler are equal. Circles 2 and 3 are identical, and the circumference of circle 1 is 20π cm. What is the radius of circle 2 in cm? | A. 2.5; B. 3; C. 2; D. 5; E. No correct answer | D | 3steps_2 | 1,277 | 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. |
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3stpes_32 | testmini | Area of Circles | As shown in the figure, a right-angled triangle and three circular paper pieces are stacked. The lengths of the two legs of the right-angled triangle are equal. The radius of circle 2 is as shown in the figure. What is the area of circle 2? ( ) cm² | A. 24π; B. 26π; C. 25π; D. No correct answer | C | 3steps_3 | 1,442 | 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. |
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3stpes_32 | testmini | Understanding Triangular Rulers;Circumference of Circles;Area of Circles | As shown in the figure, a triangular ruler is stacked with three circular paper pieces. Circles 2 and 3 are identical. It is known that the circumference of circle 1 is 20π cm. What is the area of circle 2 in cm²? | A. 24π; B. 26π; C. 25π; D. No correct answer | C | 3steps_multi | 1,607 | 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°.
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.
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. |
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3stpes_33 | testmini | Volume and Capacity of Cubes | As shown in the figure, there is an isosceles trapezoid made of wire. This wire can also be used to form a cube frame. The volume of the cube frame is shown in the figure. What is the edge length of the cube? | A. 1cm; B. 5cm; C. 2cm; D. No correct answer | A | 3steps_1 | 1,113 | Volume and Capacity of Cubes:
1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length.
2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³.
3. The volume of a cube is equal to the base area multiplied by the height.
4. The capacity of a cube usually refers to the amount of space a cubic 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. |
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3stpes_33 | testmini | Properties and Understanding of Cubes | As shown in the figure, there is an isosceles trapezoid made of wire. This piece of wire can also be bent into a square frame on the right. What is the length of the wire? ( ) cm | A. 12 cm; B. 10 cm; C. 8 cm; D. No correct answer | A | 3steps_2 | 1,278 | 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. |
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3stpes_33 | testmini | Perimeter of Trapezoids | As shown in the figure, a 12 cm long wire is bent into an isosceles trapezoid. What is the length of the legs of the trapezoid in cm? | A. 4; B. 5; C. 2; D. No correct answer | C | 3steps_3 | 1,443 | 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. |
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3stpes_33 | testmini | Volume and Capacity of Cubes;Properties and Understanding of Cubes;Perimeter of Trapezoids | As shown in the figure, there is an isosceles trapezoid made of wire. This wire can also be used to form a right-angled cube frame, and the volume of the cube frame is shown in the figure. What is the length of the legs of the isosceles trapezoid in cm? | A. 4; B. 5; C. 2; D. No correct answer | C | 3steps_multi | 1,608 | Volume and Capacity of Cubes:
1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length.
2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³.
3. The volume of a cube is equal to the base area multiplied by the height.
4. The capacity of a cube usually refers to the amount of space a cubic 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.
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.
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. |
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3stpes_34 | testmini | Properties and Understanding of Cubes | There are two iron wires of the same length, which are respectively formed into the following cube frame and rectangular cuboid frame. Based on the edge length of the cube frame, what is the length of the iron wire? ( ) | A. 60cm; B. 80cm; C. 100cm; D. No correct answer | A | 3steps_1 | 1,114 | 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. |
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3stpes_34 | testmini | Properties and Understanding of Rectangular Cuboids | There are two iron wires, each 60 cm long, which are respectively formed into the following cube frame and rectangular cuboid frame. What is the height of the rectangular cuboid? | A. 7 cm; B. 8 cm; C. 9 cm; D. No correct answer | A | 3steps_2 | 1,279 | Properties and Understanding of Rectangular Cuboids:
1. A rectangular cubiod has six faces, all of which are rectangles.
2. The opposite faces of a rectangular cubed have equal area, and of the 12 edges, the lengths of the 4 opposite edges are equal.
3. A rectangular cubiod has 8 vertices.
4. The lengths of the three edges intersecting at one vertex of a rectangular cubiod are called length, width, and height.
5. The edge where two faces of a rectangular cubiod meet is called an edge.
6. The point where three edges of a rectangular cubiod meet is called a vertex.
7. When placed on a table, a rectangular cubiod can show at most three faces.
8. The total area of the six faces of a rectangular cubiod is called its surface area. |
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3stpes_34 | testmini | Volume and Capacity of Rectangular Cuboids | There are two iron wires, each 60 cm long, which are respectively formed into the following cube frame and rectangular cuboid frame. What is the volume of the rectangular cuboid in cm³? | A. 114; B. 125; C. 105; D. No correct answer | C | 3steps_3 | 1,444 | 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. |
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3stpes_34 | testmini | Properties and Understanding of Cubes;Properties and Understanding of Rectangular Cuboids;Volume and Capacity of Rectangular Cuboids | There are two iron wires of the same length, which are respectively formed into the following cube frame and rectangular cuboid frame. What is the volume of the rectangular cuboid? ( ) cm³ | A. 114; B. 125; C. 105; D. No correct answer | C | 3steps_multi | 1,609 | 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.
Properties and Understanding of Rectangular Cuboids:
1. A rectangular cubiod has six faces, all of which are rectangles.
2. The opposite faces of a rectangular cubed have equal area, and of the 12 edges, the lengths of the 4 opposite edges are equal.
3. A rectangular cubiod has 8 vertices.
4. The lengths of the three edges intersecting at one vertex of a rectangular cubiod are called length, width, and height.
5. The edge where two faces of a rectangular cubiod meet is called an edge.
6. The point where three edges of a rectangular cubiod meet is called a vertex.
7. When placed on a table, a rectangular cubiod can show at most three faces.
8. The total area of the six faces of a rectangular cubiod is called its surface area.
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. |
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3stpes_35 | testmini | Properties and Understanding of Rectangles | As shown in the figure, rectangle ABCD is formed by a piece of wire. What is the length of side CD? | A. 36cm; B. 12cm; C. 24cm; D. No correct answer | A | 3steps_1 | 1,115 | Properties and Understanding of Rectangles:
1. A rectangle is a plane figure and a parallelogram with one right angle. A rectangle is also defined as a parallelogram with all four angles being right angles. A square is a special type of rectangle where all four sides are equal in length.
2. Properties of a rectangle: The two diagonals are equal in length; the two diagonals bisect each other; both pairs of opposite sides are parallel; both pairs of opposite sides are equal in length; all four angles are right angles; there are two axes of symmetry; it is unstable (prone to deformation); the square of the length of a rectangle's diagonal is equal to the sum of the squares of its two sides; the quadrilateral formed by sequentially connecting the midpoints of a rectangle's sides is a rhombus. |
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3stpes_35 | testmini | Properties and Understanding of Cubes | As shown in the figure, rectangle ABCD is formed by a piece of wire. If the side CD is cut off to form the cube on the right, what is the edge length of the cube? ( ) | A. 3cm; B. 4cm; C. 5cm; D. No correct answer | A | 3steps_2 | 1,280 | 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. |
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3stpes_35 | testmini | Volume and Capacity of Cubes | As shown in the figure, rectangle ABCD is formed by a piece of wire. If the side CD is cut off and used to form a cube on the right, what is the volume of the cube? ( ) cm³ | A. 24; B. 52; C. 27; D. No correct answer | C | 3steps_3 | 1,445 | Volume and Capacity of Cubes:
1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length.
2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³.
3. The volume of a cube is equal to the base area multiplied by the height.
4. The capacity of a cube usually refers to the amount of space a cubic 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. |
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3stpes_35 | testmini | Properties and Understanding of Rectangles;Properties and Understanding of Cubes;Volume and Capacity of Cubes | As shown in the figure, rectangle ABCD is formed by a piece of wire. If the side CD is cut off to form the cube on the right, what is the volume of the cube? ( ) cm³ | A. 24; B. 52; C. 27; D. No correct answer | C | 3steps_multi | 1,610 | Properties and Understanding of Rectangles:
1. A rectangle is a plane figure and a parallelogram with one right angle. A rectangle is also defined as a parallelogram with all four angles being right angles. A square is a special type of rectangle where all four sides are equal in length.
2. Properties of a rectangle: The two diagonals are equal in length; the two diagonals bisect each other; both pairs of opposite sides are parallel; both pairs of opposite sides are equal in length; all four angles are right angles; there are two axes of symmetry; it is unstable (prone to deformation); the square of the length of a rectangle's diagonal is equal to the sum of the squares of its two sides; the quadrilateral formed by sequentially connecting the midpoints of a rectangle's sides is a rhombus.
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.
Volume and Capacity of Cubes:
1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length.
2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³.
3. The volume of a cube is equal to the base area multiplied by the height.
4. The capacity of a cube usually refers to the amount of space a cubic 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. |
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3stpes_36 | testmini | Understanding Circles | As shown in the figure, a circular frame is made of wire. Based on the information in the figure, what is the length of side CD? ( ) | A. 12 cm; B. 8 cm; C. 24 cm; D. No correct answer | A | 3steps_1 | 1,116 | 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. |
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3stpes_36 | testmini | Properties and Understanding of Cubes | As shown in the figure, a circular wire frame is cut at section CD and formed into a cube. What is the edge length of the cube? ( ) | A. 1 cm; B. 2 cm; C. 3 cm; D. No correct answer | A | 3steps_2 | 1,281 | 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. |
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3stpes_36 | testmini | Volume and Capacity of Cubes | As shown in the figure, a circular wire frame is cut at segment CD and formed into a cube. What is the volume of the cube? ( ) cm³ | A. 4; B. 5; C. 1; D. No correct answer | C | 3steps_3 | 1,446 | Volume and Capacity of Cubes:
1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length.
2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³.
3. The volume of a cube is equal to the base area multiplied by the height.
4. The capacity of a cube usually refers to the amount of space a cubic 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. |
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3stpes_36 | testmini | Understanding Circles;Properties and Understanding of Cubes;Volume and Capacity of Cubes | As shown in the figure, a circular wire frame is cut at section CD and formed into a cube. What is the volume of the cube? ( ) cm³ | A. 4; B. 5; C. 1; D. No correct answer | C | 3steps_multi | 1,611 | 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 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.
Volume and Capacity of Cubes:
1. The formula for calculating the volume of a cube: The volume of a cube = side length × side length × side length.
2. In terms of letters: If V represents the volume of the cube and a represents the side length of the cube, then the formula for the volume of the cube can be written as: V = a³.
3. The volume of a cube is equal to the base area multiplied by the height.
4. The capacity of a cube usually refers to the amount of space a cubic 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. |
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3stpes_37 | testmini | Perimeter of Trapezoids | As shown in the figure, quadrilateral GBCD is an isosceles trapezoid with a perimeter of 40cm. What is the length of CD in cm? | A. 10; B. 8; C. 24; D. No correct answer | A | 3steps_1 | 1,117 | 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. |
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3stpes_37 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, quadrilateral GBCD is an isosceles trapezoid, and quadrilateral ABCD is a parallelogram. What is the length of AB in cm? | A. 10; B. 7; C. 8; D. 9; E. No correct answer | A | 3steps_2 | 1,282 | 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. |
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3stpes_37 | testmini | Area of Triangles | As shown in the figure, quadrilateral GBCD is an isosceles trapezoid. The lengths of AB and GH are given in the figure. What is the area of triangle GAB in cm²? | A. 24; B. 50; C. 40; D. No correct answer | C | 3steps_3 | 1,447 | 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. |
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3stpes_37 | testmini | Perimeter of Trapezoids;Properties and Understanding of Parallelograms;Area of Triangles | As shown in the figure, quadrilateral GBCD is an isosceles trapezoid (with a perimeter of 40 cm), and quadrilateral ABCD is a parallelogram. What is the area of triangle GAB in cm²? | A. 24; B. 50; C. 40; D. No correct answer | C | 3steps_multi | 1,612 | 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.
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. |
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3stpes_38 | testmini | Understanding Triangular Rulers | As shown in the figure, ∠4 in the right triangle is equal to ∠5 in the left triangular ruler. What is the value of ∠4? | A. 45°; B. 60°; C. 120°; D. No correct answer | A | 3steps_1 | 1,118 | 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°. |
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3stpes_38 | testmini | Sum of Interior Angles of Triangles | As shown in the figure, the left triangle is a triangular ruler. What is the measure of angle ∠1 in the right triangle? | A. 55°; B. 45°; C. 65°; D. No correct answer | A | 3steps_2 | 1,283 | 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. |
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3stpes_38 | testmini | Sum of Interior Angles of Other Polygons | In the triangle shown in the figure, the degrees of each angle are marked in the diagram. What is the value of ∠2 + ∠3 = ()°? | A. 200; B. 250; C. 260; D. No correct answer | C | 3steps_3 | 1,448 | Sum of Interior Angles of Other Polygons:
1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°.
2. The sum of the interior angles of a polygon = 180° × (number of sides - 2).
3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°.
4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon. |
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3stpes_38 | testmini | Understanding Triangular Rulers;Sum of Interior Angles of Triangles;Sum of Interior Angles of Other Polygons | As shown in the figure, ∠4 in the right triangle is equal to ∠5 in the left triangular ruler. What is ∠2 + ∠3 = ()°? | A. 200; B. 250; C. 260; D. No correct answer | C | 3steps_multi | 1,613 | 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°.
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.
Sum of Interior Angles of Other Polygons:
1. The sum of the interior angles of a quadrilateral is 360°, and the sum of the interior angles of a pentagon is 540°.
2. The sum of the interior angles of a polygon = 180° × (number of sides - 2).
3. The sum of the interior angles of an n-sided polygon = (n - 2) × 180°.
4. A polygon with all sides of equal length and all interior angles equal is called a regular polygon. |
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3stpes_39 | testmini | perpendicularity | In a parallelogram-shaped pool, the shortest path from edge AB to edge CD is 4 meters. Which of the following corresponds to this path? What is its positional relationship with line segments AB and CD? | A. ①, perpendicular; B. ②, intersecting but not perpendicular; C. ②, perpendicular; D. ③, perpendicular; E. No correct answer | C | 3steps_1 | 1,119 | perpendicularity:
1. Perpendicular lines intersect at right angles, and their intersection point is called the foot of the perpendicular. If a point on one line intersects another line forming a right angle, then this line is perpendicular to the other line.
2. A line is perpendicular to another line if the angle between them is 90 degrees. Similarly, a line is perpendicular to a plane if it forms a 90-degree angle with any line lying in that plane.
3. The perpendicular segment from a point to a line is the shortest distance from the point to the line.
4. From a given point on a line and a point not on the line, only one perpendicular line can be drawn to the given line. |
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3stpes_39 | testmini | Properties and Understanding of Parallelograms | The shortest path from side AB to side CD is path ②, and path ② is perpendicular to AB. What is the height of the parallelogram in meters? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_2 | 1,284 | 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. |
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3stpes_39 | testmini | Area of Parallelograms | The height of the parallelogram is 4m. What is the area of the parallelogram ABCD shown in the figure in square meters? | A. 12; B. 20; C. 24; D. No correct answer | C | 3steps_3 | 1,449 | Area of Parallelograms:
1. The area of a parallelogram is equal to the base times the height, S = ah. |
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3stpes_39 | testmini | perpendicularity;Properties and Understanding of Parallelograms;Area of Parallelograms | A parallelogram-shaped pool has a shortest distance of 4 meters from AB to CD. What is the area of this pool in square meters? | A. 12; B. 20; C. 24; D. No correct answer | C | 3steps_multi | 1,614 | perpendicularity:
1. Perpendicular lines intersect at right angles, and their intersection point is called the foot of the perpendicular. If a point on one line intersects another line forming a right angle, then this line is perpendicular to the other line.
2. A line is perpendicular to another line if the angle between them is 90 degrees. Similarly, a line is perpendicular to a plane if it forms a 90-degree angle with any line lying in that plane.
3. The perpendicular segment from a point to a line is the shortest distance from the point to the line.
4. From a given point on a line and a point not on the line, only one perpendicular line can be drawn to the given 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.
Area of Parallelograms:
1. The area of a parallelogram is equal to the base times the height, S = ah. |
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3stpes_40 | testmini | Perimeter of Trapezoids | As shown in the figure, the isosceles trapezoid on the left is made of wire. What is the length of the wire? ( ) cm? | A. 12; B. 10; C. 8; D. No correct answer | A | 3steps_1 | 1,120 | 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. |
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3stpes_40 | testmini | Properties and Understanding of Cubes | As shown in the figure, the isosceles trapezoid on the left is formed using a 12 cm long wire. This wire can also be used to form the three-dimensional shape on the right. What is the edge length of this three-dimensional shape in cm? | A. 1; B. 2; C. 3; D. No correct answer | A | 3steps_2 | 1,285 | 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. |
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3stpes_40 | testmini | Surface Area of Cubes | As shown in the figure, the isosceles trapezoid on the left is formed using a 12 cm long wire. This wire can also be used to form the square on the right. What is the surface area of this square in cm²? | A. 6; B. 12; C. 18; D. No correct answer | A | 3steps_3 | 1,450 | Surface Area of Cubes:
1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s.
2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6.
3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. |
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3stpes_40 | testmini | Perimeter of Trapezoids;Properties and Understanding of Cubes;Surface Area of Cubes | As shown in the figure, the isosceles trapezoid on the left is formed with a wire. This wire can also be used to form the square on the right. What is the surface area of this square in cm²? | A. 6; B. 12; C. 18; D. No correct answer | A | 3steps_multi | 1,615 | 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.
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.
Surface Area of Cubes:
1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s.
2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6.
3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. |
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3stpes_41 | testmini | Perimeter of Parallelograms | As shown in the figure, the following parallelogram is enclosed with wire. What is the length of the wire in cm? | A. 16; B. 12; C. 10; D. No correct answer | A | 3steps_1 | 1,121 | 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. |
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3stpes_41 | testmini | Properties and Understanding of Rectangular Cuboids | As shown in the figure, a 16 cm long wire can be used to form the parallelogram above. The same wire can also be used to form the rectangular cuboid below. What is the height of the rectangular cuboid in cm? | A. 1; B. 2; C. 3; D. No correct answer | A | 3steps_2 | 1,286 | Properties and Understanding of Rectangular Cuboids:
1. A rectangular cubiod has six faces, all of which are rectangles.
2. The opposite faces of a rectangular cubed have equal area, and of the 12 edges, the lengths of the 4 opposite edges are equal.
3. A rectangular cubiod has 8 vertices.
4. The lengths of the three edges intersecting at one vertex of a rectangular cubiod are called length, width, and height.
5. The edge where two faces of a rectangular cubiod meet is called an edge.
6. The point where three edges of a rectangular cubiod meet is called a vertex.
7. When placed on a table, a rectangular cubiod can show at most three faces.
8. The total area of the six faces of a rectangular cubiod is called its surface area. |
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3stpes_41 | testmini | Surface Area of Rectangular Cuboids | As shown in the figure, a 16 cm long wire can be used to form the parallelogram above. The same wire can also be used to form the cuboid below. What is the surface area of the cuboid? ( ) cm² | A. 6; B. 10; C. 12; D. No correct answer | B | 3steps_3 | 1,451 | 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). |
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3stpes_41 | testmini | Perimeter of Parallelograms;Properties and Understanding of Rectangular Cuboids;Surface Area of Rectangular Cuboids | As shown in the figure, a piece of wire can be used to form the parallelogram above. The same piece of wire can also be used to form the rectangular cuboid below. What is the surface area of the rectangular cuboid in cm²? | A. 6; B. 10; C. 12; D. No correct answer | B | 3steps_multi | 1,616 | 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.
Properties and Understanding of Rectangular Cuboids:
1. A rectangular cubiod has six faces, all of which are rectangles.
2. The opposite faces of a rectangular cubed have equal area, and of the 12 edges, the lengths of the 4 opposite edges are equal.
3. A rectangular cubiod has 8 vertices.
4. The lengths of the three edges intersecting at one vertex of a rectangular cubiod are called length, width, and height.
5. The edge where two faces of a rectangular cubiod meet is called an edge.
6. The point where three edges of a rectangular cubiod meet is called a vertex.
7. When placed on a table, a rectangular cubiod can show at most three faces.
8. The total area of the six faces of a rectangular cubiod is called its surface area.
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). |
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3stpes_42 | testmini | perpendicularity | There is a trapezoidal pool, and the distance from edge AB to edge CD is 4 meters. Which of the following corresponds to this path? What is its positional relationship with line segments AB and CD? | A. ①, perpendicular; B. ②, intersecting but not perpendicular; C. ②, perpendicular; D. ③, perpendicular; E. No correct answer | C | 3steps_1 | 1,122 | perpendicularity:
1. Perpendicular lines intersect at right angles, and their intersection point is called the foot of the perpendicular. If a point on one line intersects another line forming a right angle, then this line is perpendicular to the other line.
2. A line is perpendicular to another line if the angle between them is 90 degrees. Similarly, a line is perpendicular to a plane if it forms a 90-degree angle with any line lying in that plane.
3. The perpendicular segment from a point to a line is the shortest distance from the point to the line.
4. From a given point on a line and a point not on the line, only one perpendicular line can be drawn to the given line. |
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3stpes_42 | testmini | Properties and Understanding of Trapezoids | The shortest path from side AB to side CD is path ②, and path ② is perpendicular to AB. What is the height of the trapezoid in meters? | A. 4; B. 3; C. 2; D. 1; E. No correct answer | A | 3steps_2 | 1,287 | 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. |
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3stpes_42 | testmini | Area of Trapezoids | What is the area of the trapezoid ABCD shown in the figure in square meters? | A. 12; B. 20; C. 24; D. No correct answer | C | 3steps_3 | 1,452 | 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. |
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3stpes_42 | testmini | perpendicularity;Properties and Understanding of Trapezoids;Area of Trapezoids | A trapezoidal pool has a distance of 4 meters from AB to CD. What is the area of this pool in square meters? | A. 12; B. 20; C. 24; D. No correct answer | C | 3steps_multi | 1,617 | perpendicularity:
1. Perpendicular lines intersect at right angles, and their intersection point is called the foot of the perpendicular. If a point on one line intersects another line forming a right angle, then this line is perpendicular to the other line.
2. A line is perpendicular to another line if the angle between them is 90 degrees. Similarly, a line is perpendicular to a plane if it forms a 90-degree angle with any line lying in that plane.
3. The perpendicular segment from a point to a line is the shortest distance from the point to the line.
4. From a given point on a line and a point not on the line, only one perpendicular line can be drawn to the given line.
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.
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. |
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3stpes_43 | testmini | Area of Squares | A rectangular sheet of metal has a square cut out from each of its four corners. The side length of each square is () cm. | A. 1 cm; B. 5 cm; C. 2 cm; D. No correct answer | A | 3steps_1 | 1,123 | 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. |
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3stpes_43 | testmini | Expanded View of Rectangular Cuboids | A rectangular piece of iron sheet has a 1cm square cut out from each of its four corners. Then, it is folded to form an open-top rectangular box. What are the length, width, and height of the rectangular box? ( ) | A. 10cm 5cm 1cm; B. 7cm 12cm 1cm; C. 1cm 1cm 1cm; D. No correct answer | A | 3steps_2 | 1,288 | Expanded View of Rectangular Cuboids:
1. The expanded view of a rectangular cubiod is a two-dimensional figure obtained by unfolding the faces of the rectangular cubiod and laying them flat.
2. A rectangular cubiod has 6 faces, and the expanded view consists of these 6 faces. Each face is a rectangle. Opposite faces of the rectangular cubiod are equal, so the expanded view will have three pairs of equal rectangles. |
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3stpes_43 | testmini | Surface Area of Rectangular Cuboids | A rectangular piece of iron sheet has a square cut out from each of its four corners. Then, it is folded to form an open-top rectangular cuboid as shown in the figure. What is the base area of the rectangular cuboid? () | A. 50 cm²; B. 100 cm²; C. 10 cm²; D. No correct answer | A | 3steps_3 | 1,453 | 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). |
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3stpes_43 | testmini | Area of Squares;Expanded View of Rectangular Cuboids;Surface Area of Rectangular Cuboids | A rectangular sheet of metal has a square with an area of 1 cm² cut out from each of its four corners. It is then folded into an open-top rectangular box. What is the area of the base of the box in cm²? | A. 50 cm²; B. 100 cm²; C. 10 cm²; D. No correct answer | A | 3steps_multi | 1,618 | 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.
Expanded View of Rectangular Cuboids:
1. The expanded view of a rectangular cubiod is a two-dimensional figure obtained by unfolding the faces of the rectangular cubiod and laying them flat.
2. A rectangular cubiod has 6 faces, and the expanded view consists of these 6 faces. Each face is a rectangle. Opposite faces of the rectangular cubiod are equal, so the expanded view will have three pairs of equal rectangles.
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). |
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3stpes_44 | testmini | Area of Squares | A square metal sheet has a small square cut out from each of its four corners. What is the side length of each small square in cm? | A. 1 cm; B. 5 cm; C. 3 cm; D. 2 cm; E. No correct answer | D | 3steps_1 | 1,124 | 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. |
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3stpes_44 | testmini | Expanded View of Cubes | A square metal sheet has a square with a side length of 2 cm cut off from each of its four corners. Then, it is folded to form an open-top cube. What is the edge length of the cube? ( ) cm | A. 1 cm; B. 2 cm; C. 3 cm; D. No correct answer | B | 3steps_2 | 1,289 | Expanded View of Cubes:
1. An expanded view of a cube is a two-dimensional representation of the cube, formed by unfolding the cube along its edges into a plane. It shows the arrangement of all six faces of the cube on the same plane.
2. An expanded view of a cube can have different arrangements, but it usually consists of one central square surrounded by four adjacent squares, with an additional square connected to the central square. Common shapes include "T" shape, "cross" shape, and "L" shape.
3. Each square in the expanded view represents one face of the cube, and every two adjacent squares in the expanded view are also adjacent faces in the cube.
4. There are a total of 6 squares in the expanded view, and each square has equal side lengths. |
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3stpes_44 | testmini | Surface Area of Cubes | A square metal sheet has a small square with a side length of 2 cm cut out from each of its four corners. Then, it is folded to form an open-top cube. What is the surface area of the cube? ( ) cm² | A. 8 cm²; B. 16 cm²; C. 24 cm²; D. 32 cm²; E. No correct answer | C | 3steps_3 | 1,454 | Surface Area of Cubes:
1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s.
2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6.
3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. |
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3stpes_44 | testmini | Area of Squares;Expanded View of Cubes;Surface Area of Cubes | A square piece of metal has a small square cut out from each of its four corners. It is then folded to form an open-top cube. The surface area of the cube is () cm². | A. 8 cm²; B. 16 cm²; C. 24 cm²; D. 32 cm²; E. No correct answer | C | 3steps_multi | 1,619 | 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.
Expanded View of Cubes:
1. An expanded view of a cube is a two-dimensional representation of the cube, formed by unfolding the cube along its edges into a plane. It shows the arrangement of all six faces of the cube on the same plane.
2. An expanded view of a cube can have different arrangements, but it usually consists of one central square surrounded by four adjacent squares, with an additional square connected to the central square. Common shapes include "T" shape, "cross" shape, and "L" shape.
3. Each square in the expanded view represents one face of the cube, and every two adjacent squares in the expanded view are also adjacent faces in the cube.
4. There are a total of 6 squares in the expanded view, and each square has equal side lengths.
Surface Area of Cubes:
1. Surface Area of a Cube: The total area of the 6 faces of a cube is its surface area, S = 6s.
2. Method of Calculating the Surface Area of a Cube: Surface area of a cube = edge length × edge length × 6.
3. S = 6*a*a, where S represents the surface area of the cube, and a represents the edge length of the cube. |
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3stpes_45 | testmini | Area of Triangles | As shown in the figure, the area of triangle ABE is 4 cm². What is the length of AE? ( ) cm | A. 4; B. 5; C. 2; D. No correct answer | A | 3steps_1 | 1,125 | 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. |
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3stpes_45 | testmini | Area of Trapezoids | As shown in the figure, the area of quadrilateral AECF is 14 cm². What is the length of CE? ( ) cm | A. 4; B. 5; C. 2; D. No correct answer | A | 3steps_2 | 1,290 | 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. |
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3stpes_45 | testmini | Area of Parallelograms | The side lengths of parallelogram ABCD are shown in the diagram. What is its area in cm²? | A. 24; B. 16; C. 8; D. No correct answer | A | 3steps_3 | 1,455 | Area of Parallelograms:
1. The area of a parallelogram is equal to the base times the height, S = ah. |
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3stpes_45 | testmini | Area of Triangles;Area of Trapezoids;Area of Parallelograms | As shown in the figure, the area of triangle ABE is 4 cm², and the area of trapezoid AECF is 14 cm². What is the area of parallelogram ABCD? | A. 24; B. 16; C. 8; D. No correct answer | A | 3steps_multi | 1,620 | 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 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. |
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3stpes_46 | testmini | Properties and Understanding of Trapezoids | In the isosceles trapezoid ABCD as shown in the figure, what is the length of AC? ( ) cm | A. 3; B. 4; C. 5; D. No correct answer | A | 3steps_1 | 1,126 | 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. |
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3stpes_46 | testmini | Properties and Understanding of Parallelograms | As shown in the figure, there is an isosceles trapezoid ABCD with a parallelogram ABCE inside it. What are the lengths of sides BE and CE, respectively? ( ) cm | A. 3, 5; B. 1, 4; C. 1, 6; D. 1, 8; E. No correct answer | A | 3steps_2 | 1,291 | 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. |
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3stpes_46 | testmini | Perimeter of Triangles | As shown in the figure, there is an isosceles trapezoid ABCD with a triangle BED inside it. What is the perimeter of the triangle? ( ) cm | A. 8; B. 9; C. 6; D. 7; E. No correct answer | A | 3steps_3 | 1,456 | Perimeter of Triangles:
1. The perimeter of a triangle is the sum of its three sides. Scalene triangle: C = a + b + c (where a, b, and c are the lengths of the three sides of the triangle).
2. Isosceles triangle: C = 2a + b (where a is the length of the equal sides, and b is the length of the base).
3. Equilateral triangle: C = 3a (where a is the length of any one side). |
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3stpes_46 | testmini | Properties and Understanding of Trapezoids;Properties and Understanding of Parallelograms;Perimeter of Triangles | As shown in the figure, an isosceles trapezoid is divided into a parallelogram and a triangle. The perimeter of the triangle is () centimeters. | A. 24; B. 20; C. 18; D. 8; E. No correct answer | D | 3steps_multi | 1,621 | 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.
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.
Perimeter of Triangles:
1. The perimeter of a triangle is the sum of its three sides. Scalene triangle: C = a + b + c (where a, b, and c are the lengths of the three sides of the triangle).
2. Isosceles triangle: C = 2a + b (where a is the length of the equal sides, and b is the length of the base).
3. Equilateral triangle: C = 3a (where a is the length of any one side). |
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3stpes_47 | testmini | Understanding Length (Using a Ruler) | As shown in the figure, there is a bottle in an upright position and an inverted position. What are the liquid level height when upright and the air height when inverted, respectively? | A. 6cm, 2cm; B. 1cm, 3cm; C. 1cm, 4cm; D. No correct answer | A | 3steps_1 | 1,127 | Understanding Length (Using a Ruler):
1. Length is usually measured using a ruler.
2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters.
3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end.
4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends. |
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3stpes_47 | testmini | Volume and Capacity of Cylinders | As shown in the figure, there is a bottle in an upright position and an inverted position. What are the volumes of the liquid surface when upright and the air volume when inverted, respectively, in cm³? | A. 54π, 18π; B. 60π, 18π; C. 70π, 20π; D. Cannot be determined; E. No correct answer | A | 3steps_2 | 1,292 | 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. |
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3stpes_47 | testmini | Combining and Dividing Solids | The diagram shows an inverted irregular bottle, where the volume of water is v2 and the volume of air is v1 as shown. What is the total volume of the bottle in cm³? | A. 72π; B. 78π; C. 54π; D. No correct answer | A | 3steps_3 | 1,457 | 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. |
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3stpes_47 | testmini | Understanding Length (Using a Ruler);Volume and Capacity of Cylinders;Combining and Dividing Solids | As shown in the figure, use a ruler to measure the height of the water when the bottle is upright and the height of the air when the bottle is upside down. What is the volume of the bottle in cm³? | A. 108π; B. 72π; C. 54π; D. No correct answer | B | 3steps_multi | 1,622 | Understanding Length (Using a Ruler):
1. Length is usually measured using a ruler.
2. When using a ruler for measurement, first determine the measurement unit of the ruler. If there is no special indication, the default unit is centimeters.
3. If one end of the line segment being measured is aligned with the 0 mark on the ruler, the length of the line segment is the direct reading from the other end.
4. If one end of the line segment being measured is not aligned with the 0 mark, the length of the line segment is the difference between the readings at both ends.
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.
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. |
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3stpes_48 | testmini | Area of Rectangles | As shown in the figure, the area of the rectangle below is 48 cm². What is the length of AB in the rectangle? | A. 8 cm; B. 6 cm; C. 10 cm; D. No correct answer | A | 3steps_1 | 1,128 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
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3stpes_48 | testmini | Properties of Cylinders | As shown in the figure, if the rectangle is rotated around the line along AB in the rectangle, what are the height and the radius of the base of the resulting cylinder, respectively? | A. 8cm-6cm; B. 8cm-3cm; C. 4cm-6cm; D. No correct answer | A | 3steps_2 | 1,293 | 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. |
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3stpes_48 | testmini | Volume and Capacity of Cylinders | What is the volume of the cylinder shown in the figure ( ) cm³? | A. 288π; B. 280π; C. 100π; D. No correct answer | A | 3steps_3 | 1,458 | 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. |
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3stpes_48 | testmini | Area of Rectangles;Properties of Cylinders;Volume and Capacity of Cylinders | As shown in the figure, the area of the rectangle below is 48 cm². If it is rotated one full turn around AB, a three-dimensional shape is formed. What is its volume ( ) cm³? | A. 288π; B. 280π; C. 100π; D. No correct answer | A | 3steps_multi | 1,623 | 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.
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. |
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3stpes_49 | testmini | Area of Rectangles | As shown in the figure, the area of the rectangle below is 10 cm². What is the width of the rectangle? | A. 2 cm; B. 6 cm; C. 10 cm; D. No correct answer | A | 3steps_1 | 1,129 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
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3stpes_49 | testmini | Properties of Cylinders | In the diagram, what is the 3D shape formed by rotating the rectangle around the line L for one full turn? | A. A cylinder with a height of 5 cm and a base radius of 2 cm; B. A cylinder with a height of 2 cm and a base radius of 5 cm; C. A cylinder with a height of 5 cm and a base radius of 4 cm; D. No correct answer | A | 3steps_2 | 1,294 | 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. |
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3stpes_49 | testmini | Surface Area of Cylinders | As shown in the figure, what is the base area of the cylinder in cm²?(π = 3.14) | A. 12.56; B. 1.56; C. 78.5; D. No correct answer | A | 3steps_3 | 1,459 | 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. |
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3stpes_49 | testmini | Area of Rectangles;Properties of Cylinders;Surface Area of Cylinders | As shown in the figure, a rectangle with an area of 10 cm² is rotated around the line L to form a three-dimensional shape. What is the base area of this three-dimensional shape in cm²?(π = 3.14) | A. 12.56; B. 1.56; C. 78.5; D. No correct answer | A | 3steps_multi | 1,624 | 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. |
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3stpes_50 | testmini | Representing Positions Using Ordered Pairs | The position of point A in the grid is shown in the figure. If point B can be represented by (1, 2), then point A can be represented by the coordinates (). | A. (4, 5); B. (7, 2); C. (2, 2); D. No correct answer | A | 3steps_1 | 1,130 | Representing Positions Using Ordered Pairs:
1. Definition of an Ordered Pair: An ordered pair is a combination of two numbers, typically used to represent the position of a point in a Cartesian coordinate system. The two numbers in the ordered pair represent values on different directions or axes.
2. Composition of an Ordered Pair: In an ordered pair (x, y), the first number x represents the column number, which is the horizontal position; the second number y represents the row number, which is the vertical position. This notation follows the "horizontal first, then vertical" convention.
3. Notation of an Ordered Pair: When using an ordered pair to represent a point's position, the numbers or letters representing the column and row need to be enclosed in parentheses, for example, (2, 5) represents the position at column 2, row 5.
4. Uniqueness of an Ordered Pair: Each ordered pair uniquely determines a position, and conversely, a position can be uniquely represented by an ordered pair. |
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3stpes_50 | testmini | Translation | If triangle ABC is translated two units down and then one unit to the right to obtain the new triangle A'B'C', what are the coordinates of point A' after the translation? | A. (2, 3); B. (4, 4); C. (3, 3); D. (5, 3); E. No correct answer | D | 3steps_2 | 1,295 | Translation:
1. Translation refers to the movement of all points of a figure in the same plane by the same distance in a given direction. This type of motion is called translation of the figure.
2. Translation does not change the shape, size, or orientation of the figure.
3. The shape and size of the figure remain unchanged after translation; only the position changes.
4. After translation, the line segments connecting corresponding points are parallel (or collinear) and equal in length.
5. Multiple consecutive translations are equivalent to a single translation.
6. A figure after an even number of symmetries is equivalent to the figure after a translation.
7. Translation is determined by direction and distance.
8. After translation, corresponding line segments are parallel (or collinear) and equal in length, corresponding angles are equal, and the line segments connecting corresponding points are parallel (or collinear) and equal in length. |
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3stpes_50 | testmini | Rotation | After rotating triangle ABC 90 degrees clockwise around point C, the coordinates of point A' are? | A. (11,3); B. (8,4); C. (8,3); D. (12,3); E. No correct answer | A | 3steps_3 | 1,460 | 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. |
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3stpes_50 | testmini | Representing Positions Using Ordered Pairs;Translation;Rotation | As shown in the figure below, triangle ABC is first translated 2 units downwards and then 1 unit to the right to obtain the new triangle A'B'C'. Then, point A'C' is rotated 90° clockwise around point C'. After the rotation, the coordinates of point A' are ( ). | A. (11，3); B. (8，4); C. (8，3); D. (12，3); E. No correct answer | A | 3steps_multi | 1,625 | Representing Positions Using Ordered Pairs:
1. Definition of an Ordered Pair: An ordered pair is a combination of two numbers, typically used to represent the position of a point in a Cartesian coordinate system. The two numbers in the ordered pair represent values on different directions or axes.
2. Composition of an Ordered Pair: In an ordered pair (x, y), the first number x represents the column number, which is the horizontal position; the second number y represents the row number, which is the vertical position. This notation follows the "horizontal first, then vertical" convention.
3. Notation of an Ordered Pair: When using an ordered pair to represent a point's position, the numbers or letters representing the column and row need to be enclosed in parentheses, for example, (2, 5) represents the position at column 2, row 5.
4. Uniqueness of an Ordered Pair: Each ordered pair uniquely determines a position, and conversely, a position can be uniquely represented by an ordered pair.
Translation:
1. Translation refers to the movement of all points of a figure in the same plane by the same distance in a given direction. This type of motion is called translation of the figure.
2. Translation does not change the shape, size, or orientation of the figure.
3. The shape and size of the figure remain unchanged after translation; only the position changes.
4. After translation, the line segments connecting corresponding points are parallel (or collinear) and equal in length.
5. Multiple consecutive translations are equivalent to a single translation.
6. A figure after an even number of symmetries is equivalent to the figure after a translation.
7. Translation is determined by direction and distance.
8. After translation, corresponding line segments are parallel (or collinear) and equal in length, corresponding angles are equal, and the line segments connecting corresponding points are parallel (or collinear) and equal in length.
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. |