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2steps_347 | testmini | Volume and Capacity of Rectangular Cuboids;Volume and Capacity of Cubes | As shown in the diagram, when measuring an angle with a protractor, the radius of the protractor is as shown. What is the area of the sector formed by this angle and the edge of the protractor in cm²? | A. 3π; B. 1π; C. 5π; D. No correct answer | A | 2steps_multi | 1,067 | 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.
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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2steps_348 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, measure the angle using a protractor. ∠D=( )? | A. 30°; B. 60°; C. 55°; D. 90°; E. No correct answer | B | 2steps_1 | 348 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_348 | testmini | Properties and Understanding of Parallelograms | As shown in the diagram, using a protractor to measure the angle, the degree of the angle is as shown. What is the degree of ∠B in the parallelogram? | A. 60; B. 90; C. 30; D. No correct answer | A | 2steps_2 | 708 | 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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2steps_348 | testmini | Understanding Angles (Using a Protractor);Properties and Understanding of Parallelograms | As shown in the diagram, using a protractor to measure the angle, what is the size of ∠B in the parallelogram (in degrees)? | A. 60; B. 90; C. 30; D. No correct answer | A | 2steps_multi | 1,068 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
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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2steps_349 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, using a protractor to measure the angle, what is the size of ∠1? () | A. 30°; B. 60°; C. 55°; D. 90°; E. No correct answer | A | 2steps_1 | 349 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_349 | testmini | Area of Sectors | As shown in the diagram, when measuring an angle with a protractor, the area of the sector formed by the angle being measured and the edge of the protractor is ( ) cm². | A. 3π; B. 1π; C. 5π; D. No correct answer | A | 2steps_2 | 709 | Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14. |
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2steps_349 | testmini | Understanding Angles (Using a Protractor);Area of Sectors | As shown in the diagram, when measuring an angle with a protractor, the radius of the protractor is 6 cm. What is the area of the sector formed by this angle and the edge of the protractor in cm²? | A. 3π; B. 1π; C. 5π; D. No correct answer | A | 2steps_multi | 1,069 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
Area of Sectors:
1. Since the area of a sector with a central angle of 360° is the area of the circle, S = πr², the area of a sector with a central angle of n° is: S = nπr² ÷ 360.
2. There is another formula for the area of a sector: S = 1/2 lr, where l is the arc length and r is the radius. The arc length l = nπr ÷ 180
3. Generally, π is taken as 3.14. |
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2steps_350 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, ∠BAC=( )? | A. 30°; B. 60°; C. 45°; D. 90°; E. No correct answer | C | 2steps_1 | 350 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_350 | testmini | Southeast, Southwest, Northeast, Northwest Directions | As shown in the diagram, the degree of ∠BAC is as shown. The point located to the northwest of point A is () | A. Point B; B. Point C; C. Point D; D. No correct answer | A | 2steps_2 | 710 | Southeast, Southwest, Northeast, Northwest Directions:
1.Northeast lies 45 degrees north of due east, southeast lies 45 degrees south of due east, northwest lies 45 degrees west of due north, and southwest lies 45 degrees west of due south. |
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2steps_350 | testmini | Volume and Capacity of Rectangular Cuboids;Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters) | As shown in the diagram, there are several points around the protractor. Which point is located to the northwest of point A? | A. Point B; B. Point C; C. Point D; D. No correct answer | A | 2steps_multi | 1,070 | 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.
Conversion Rates and Calculations Between Volume Units (Including Liters and Milliliters):
1. Conversion rates between volume units: 1 cubic meter (m³) = 1000 cubic decimeters (dm³); 1 cubic decimeter (dm³) = 1000 cubic centimeters (cm³); 1 cubic decimeter (dm³) = 1 liter (L); 1 cubic centimeter (cm³) = 1 milliliter (mL)
2. Converting between Different Units: To convert from a larger unit to a smaller unit, multiply by the conversion rate. To convert from a smaller unit to a larger unit, divide by the conversion rate.
3. To convert cubic meters to cubic decimeters: multiply by 1000.
4. To convert cubic decimeters to cubic centimeters: multiply by 1000.
5. To convert cubic decimeters to liters: 1 cubic decimeter equals 1 liter.
6. To convert cubic centimeters to milliliters: 1 cubic centimeter equals 1 milliliter.
7. To convert milliliters to cubic centimeters: 1 milliliter equals 1 cubic centimeter.
8. To convert liters to cubic decimeters: 1 liter equals 1 cubic decimeter.
9. To convert cubic decimeters to cubic meters: divide by 1000.
10. To convert cubic centimeters to cubic decimeters: divide by 1000.
11. To convert cubic decimeters to cubic meters: divide by 1000. |
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2steps_351 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, measure the angle with a protractor. ∠D=( )? | A. 30°; B. 60°; C. 45°; D. 90°; E. No correct answer | C | 2steps_1 | 351 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_351 | testmini | Properties and Understanding of Trapezoids | As shown in the diagram, using a protractor to measure the angle, what is the size of ∠C in the isosceles trapezoid? | A. 135°; B. 45°; C. 90°; D. 60°; E. No correct answer | B | 2steps_2 | 711 | 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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2steps_351 | testmini | Understanding Angles (Using a Protractor);Properties and Understanding of Trapezoids | As shown in the diagram, what is the size of angle C in the isosceles trapezoid? | A. 135°; B. 45°; C. 90°; D. 60°; E. No correct answer | B | 2steps_multi | 1,071 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
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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2steps_352 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, measure the angle with a protractor. ∠D=( )? | A. 30°; B. 60°; C. 45°; D. 90°; E. No correct answer | B | 2steps_1 | 352 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_352 | testmini | Properties and Understanding of Trapezoids | As shown in the diagram, using a protractor to measure angles, ∠D=60°, what is the size of angle ∠C in the isosceles trapezoid? | A. 120; B. 100; C. 60; D. No correct answer | C | 2steps_2 | 712 | 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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2steps_352 | testmini | Understanding Angles (Using a Protractor);Properties and Understanding of Trapezoids | As shown in the diagram, when measuring the angle with a protractor, what is the size of angle ∠C in the isosceles trapezoid? | A. 120°; B. 100°; C. 60°; D. No correct answer | C | 2steps_multi | 1,072 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
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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2steps_353 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, using a protractor to measure the angle, what is the degree of ∠XOA? () | A. 30°; B. 60°; C. 45°; D. 90°; E. No correct answer | B | 2steps_1 | 353 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_353 | testmini | Determining the Positions of Objects Based on Direction, Angle, and Distance | As shown in the diagram, when measuring angles with a protractor, the point at 60° east by north from point O is ( )? | A. Point A; B. Point B; C. Point C; D. No correct answer | A | 2steps_2 | 713 | Determining the Positions of Objects Based on Direction, Angle, and Distance:
1. Determining Direction: Common basic directions include east, west, south, and north, as well as intermediate directions such as southeast, southwest, northeast, and northwest. In mathematics, "up" usually represents north, "down" represents south, "left" represents west, and "right" represents east to indicate directions.
2. Determining Angle: Angles are used to describe the relative position between objects or the deviation of an object from a standard direction. For example, "30 degrees west of north" describes a direction that is 30 degrees to the left of the north direction.
3. Determining Distance: Distance refers to the straight-line length from one point to another. When describing a position, it is necessary to specify the distance along a certain direction from a reference point.
4. Describing Position: To describe the position of an object, first determine a reference point, then describe the direction, angle, and distance from the reference point to the object based on its actual location. For example, "The hospital is located 100 meters in the direction 30 degrees west of north from the library." |
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2steps_353 | testmini | Understanding Angles (Using a Protractor);Determining the Positions of Objects Based on Direction, Angle, and Distance | As shown in the diagram, when measuring angles with a protractor, the point at 60° east by north from point O is ( )? | A. Point A; B. Point B; C. Point C; D. No correct answer | A | 2steps_multi | 1,073 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
Determining the Positions of Objects Based on Direction, Angle, and Distance:
1. Determining Direction: Common basic directions include east, west, south, and north, as well as intermediate directions such as southeast, southwest, northeast, and northwest. In mathematics, "up" usually represents north, "down" represents south, "left" represents west, and "right" represents east to indicate directions.
2. Determining Angle: Angles are used to describe the relative position between objects or the deviation of an object from a standard direction. For example, "30 degrees west of north" describes a direction that is 30 degrees to the left of the north direction.
3. Determining Distance: Distance refers to the straight-line length from one point to another. When describing a position, it is necessary to specify the distance along a certain direction from a reference point.
4. Describing Position: To describe the position of an object, first determine a reference point, then describe the direction, angle, and distance from the reference point to the object based on its actual location. For example, "The hospital is located 100 meters in the direction 30 degrees west of north from the library." |
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2steps_354 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, measure the angle with a protractor, ∠C = （） | A. 30°; B. 60°; C. 55°; D. 90°; E. No correct answer | B | 2steps_1 | 354 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_354 | testmini | Properties and Understanding of Trapezoids | As shown in the diagram, using a protractor to measure the angle, ∠C=60°, then the length of CD in the trapezoid is ( ) cm. | A. 6; B. 8; C. 10; D. No correct answer | A | 2steps_2 | 714 | 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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2steps_354 | testmini | Understanding Angles (Using a Protractor);Properties and Understanding of Trapezoids | As shown in the diagram, the length of CD in the trapezoid is ( ) cm. | A. 6; B. 8; C. 10; D. No correct answer | A | 2steps_multi | 1,074 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
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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2steps_355 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, triangle ABC is a cross-section of a cone. What is the measure of ∠1? | A. 30°; B. 45°; C. 60°; D. 90°; E. No correct answer | B | 2steps_1 | 355 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_355 | testmini | Properties of Cones | As shown in the diagram, triangle ABC is a cross-section of a cone. It is known that ∠1 = 45°. What is the measure of ∠2? | A. 30°; B. 45°; C. 60°; D. 90°; E. No correct answer | B | 2steps_2 | 715 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
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2steps_355 | testmini | Understanding Angles (Using a Protractor);Properties of Cones | As shown in the diagram, triangle ABC is a cross-section of a cone. What is the measure of ∠2? | A. 30°; B. 45°; C. 60°; D. 90°; E. No correct answer | B | 2steps_multi | 1,075 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
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2steps_356 | testmini | Understanding Angles (Using a Protractor) | As shown in the diagram, Mike is measuring the degree of an angle. Based on the measurement results shown in the diagram, this angle is ( )°. | A. 90; B. 92; C. 89; D. Cannot be determined; E. No correct answer | C | 2steps_1 | 356 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle. |
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2steps_356 | testmini | Understanding and Representing Angles | As shown in the diagram, this angle is ( ) | A. Right angle; B. Acute angle; C. Obtuse angle; D. Straight angle; E. No correct answer | B | 2steps_2 | 716 | Understanding and Representing Angles:
1. Understanding angles from a static perspective: An angle is a figure formed by two rays originating from a single point.
2. Understanding angles from a dynamic perspective: When a ray rotates around its vertex to another position, the figure formed by these two rays is called an angle. Two rays with a common endpoint form an angle, this common endpoint is called the vertex of the angle, and the two rays are called the sides of the angle.
3. Since rays extend infinitely in one direction, the length of the sides of an angle is irrelevant to the size of the angle.
4. The size of an angle can be measured and compared.
5. Straight angle: A 180° angle. When the two sides of an angle are on the same line, the angle formed is called a straight angle. Specifically, when the ray OA rotates around point O, and the terminal side is on the extension line of the initial side OA in the opposite direction, it forms a straight angle.
6. Right angle: A 90° angle. When the ray OA rotates around point O, and the terminal side is perpendicular to the initial side, it forms a right angle. Half of a straight angle is called a right angle.
7. Acute angle: An angle greater than 0° and less than 90°. An angle smaller than a right angle is called an acute angle.
8. Obtuse angle: An angle greater than 90° and less than 180°. An angle greater than a right angle and less than a straight angle is called an obtuse angle.
9. Full angle: A 360° angle. When the ray OA rotates around point O, and the terminal side coincides with the initial side, it forms a full angle. |
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2steps_356 | testmini | Understanding Angles (Using a Protractor);Understanding and Representing Angles | As shown in the diagram, Mike is measuring the degree of an angle. Based on the measurement results shown in the diagram, this angle is a ( ). | A. Right angle; B. Acute angle; C. Obtuse angle; D. Straight angle; E. No correct answer | B | 2steps_multi | 1,076 | Understanding Angles (Using a Protractor):
1. Steps to measure an angle: Align the center of the protractor with the vertex of the angle, and align the 0° baseline of the protractor with one side of the angle. The degree measure of the angle is indicated by the other side of the angle on the protractor. When reading the degree measure, pay attention to whether to use the outer scale or the inner scale. If the angle opens to the left, use the outer scale; if the angle opens to the right, use the inner scale.
2. The size of an angle is not related to the length of its two sides but to the amount they open. The wider the two sides open, the larger the angle; the narrower they open, the smaller the angle. Full angle > straight angle > obtuse angle > right angle > acute angle.
Understanding and Representing Angles:
1. Understanding angles from a static perspective: An angle is a figure formed by two rays originating from a single point.
2. Understanding angles from a dynamic perspective: When a ray rotates around its vertex to another position, the figure formed by these two rays is called an angle. Two rays with a common endpoint form an angle, this common endpoint is called the vertex of the angle, and the two rays are called the sides of the angle.
3. Since rays extend infinitely in one direction, the length of the sides of an angle is irrelevant to the size of the angle.
4. The size of an angle can be measured and compared.
5. Straight angle: A 180° angle. When the two sides of an angle are on the same line, the angle formed is called a straight angle. Specifically, when the ray OA rotates around point O, and the terminal side is on the extension line of the initial side OA in the opposite direction, it forms a straight angle.
6. Right angle: A 90° angle. When the ray OA rotates around point O, and the terminal side is perpendicular to the initial side, it forms a right angle. Half of a straight angle is called a right angle.
7. Acute angle: An angle greater than 0° and less than 90°. An angle smaller than a right angle is called an acute angle.
8. Obtuse angle: An angle greater than 90° and less than 180°. An angle greater than a right angle and less than a straight angle is called an obtuse angle.
9. Full angle: A 360° angle. When the ray OA rotates around point O, and the terminal side coincides with the initial side, it forms a full angle. |
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2steps_357 | testmini | Area of Circles | What is the area of the circle in the diagram below (in square centimeters)?(π = 3.14) | A. 314; B. 628; C. 320.28; D. No correct answer | A | 2steps_1 | 357 | 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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2steps_357 | testmini | Area of Rectangles | In the diagram below, the area of the rectangle is 314 square centimeters. The width of the rectangle is ( ) cm.(π = 3.14) | A. 3.14; B. 6.28; C. 15.7; D. No correct answer | C | 2steps_2 | 717 | Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
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2steps_357 | testmini | Area of Circles;Area of Rectangles | The areas of the two shapes below are equal. The width of the rectangle is ( ) cm.(π = 3.14) | A. 3.14; B. 6.28; C. 15.7; D. No correct answer | C | 2steps_multi | 1,077 | Area of Circles:
1. The area of a circle = pi × radius × radius. S = πr² = π( d/2 )²
2. Generally, π is taken as 3.14.
Area of Rectangles:
1. The area of a rectangle is equal to its length × width, expressed as: S = ab. |
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2steps_358 | testmini | Properties of Cones | As shown in the diagram, what object is formed by rotating triangle AB around its side as an axis for one complete turn, and what are the radius and height of the base? | A. Cone - 3cm - 4cm; B. Cone - 4cm - 3cm; C. Cylinder - 3cm - 4cm; D. Cylinder - 4cm - 3cm; E. No correct answer | A | 2steps_1 | 358 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length. |
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2steps_358 | testmini | Volume and Capacity of Cones | As shown in the diagram, what is the volume of the cone in cm³? (π is taken as 3.14) | A. 9.42; B. 113.04; C. 37.68; D. 12.56; E. No correct answer | C | 2steps_2 | 718 | Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
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2steps_358 | testmini | Properties of Cones;Volume and Capacity of Cones | As shown in the diagram, rotating a triangle around side AB for one full turn forms a solid. The volume of this solid is ( ) cm³.(π = 3.14) | A. 9.42; B. 113.04; C. 37.68; D. 12.56; E. No correct answer | C | 2steps_multi | 1,078 | Properties of Cones:
1. The base of a cone is a circle, and the lateral surface of a cone is a curved surface.
2. The distance from the apex of the cone to the center of the base is the height of the cone.
3. When the lateral surface of a cone is unfolded, it forms a sector.
4. The line segment from the apex of the cone to any point on the edge of the base is called the slant height of the cone, and all slant heights are equal in length.
Volume and Capacity of Cones:
1. Volume formula: The formula for calculating the volume of a cone is V = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone. This formula indicates that the volume of a cone is one-third the volume of a cylinder with the same base and height.
2. Relationship between height and volume: If the volume of a cone is known, the height of the cone can be calculated using the formula h = 3V/(πr²), where V is the volume of the cone and r is the radius of the base.
3. Relationship between base area and volume: Similarly, if the volume of a cone is known, the base area of the cone can be calculated using the formula A = 3V/h, where V is the volume of the cone and h is the height.
4. The capacity of a cone usually refers to the amount of space a conical container can hold. In the absence of special instructions, capacity and volume can be considered the same concept, using the same units and calculation methods.
5. Generally, π is taken as 3.14. |
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2steps_359 | testmini | Folding Problems of Figures | After folding a circle multiple times and then unfolding it, the creases obtained are all the circle's ( )? | A. Radius; B. Diameter; C. Cannot be determined; D. No correct answer | B | 2steps_1 | 359 | Folding Problems of Figures:
1. The folded figure exhibits the properties of a symmetrical figure, with the fold line as the axis of symmetry, and the two parts of the figure before and after folding are symmetrical about the fold line. |
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2steps_359 | testmini | Understanding Circles | Are all diameters of a circle equal in length? | A. Equal; B. Not equal; C. Cannot be determined; D. No correct answer | A | 2steps_2 | 719 | 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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2steps_359 | testmini | Folding Problems of Figures;Understanding Circles | As shown in the diagram, a circular paper is folded multiple times and then unfolded, resulting in several creases. Are the lengths of each crease equal? | A. Equal; B. Not equal; C. Cannot be determined; D. No correct answer | A | 2steps_multi | 1,079 | Folding Problems of Figures:
1. The folded figure exhibits the properties of a symmetrical figure, with the fold line as the axis of symmetry, and the two parts of the figure before and after folding are symmetrical about the fold line.
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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2steps_360 | testmini | Expanded View of Cylinders | As shown in the diagram, it is known that this figure is the lateral surface development of a cylinder. What is the circumference of the base of the cylinder?(π = 3.14) | A. 1 cm; B. 1.5 cm; C. 3.14 cm; D. 6.28 cm; E. No correct answer | D | 2steps_1 | 360 | 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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2steps_360 | testmini | Circumference of Circles | As shown in the diagram, it is known that this figure is the lateral surface development of a cylinder. If the circumference of the cylinder's base is 6.28 cm, what is the radius of the base?(π = 3.14) | A. 1 cm; B. 1.5 cm; C. 3.14 cm; D. 6.28 cm; E. No correct answer | A | 2steps_2 | 720 | 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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2steps_360 | testmini | Expanded View of Cylinders;Circumference of Circles | As shown in the diagram, it is known that this figure is the lateral surface development of a cylinder. What is the radius of the cylinder's base? (π = 3.14) | A. 1 cm; B. 1.5 cm; C. 3.14 cm; D. 6.28 cm; E. No correct answer | A | 2steps_multi | 1,080 | 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. |