Geometry & Trigonometry practice questions

Geometry & Trigonometry questions

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57. 57CirclesStandard

    A circle has a circumference of 36π. Enter its radius.A circle has a circumference of 36π. Enter its radius.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 1818

    Circumference is 2π r, so 2π r = 36π gives r = 18. Using the area formula instead is the usual wrong turn here.Circumference is 2π r, so 2π r = 36π gives r = 18. Using the area formula instead is the usual wrong turn here.

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58. 58VolumeStandard

    A cylinder has radius 5 cm and height 12 cm. What is its volume, in terms of π?A cylinder has radius 5 cm and height 12 cm. What is its volume, in terms of π?

    • A300π cubic cm300π cubic cm
    • B120π cubic cm120π cubic cm
    • C60π cubic cm60π cubic cm
    • D900π cubic cm900π cubic cm
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    Answer: 300π cubic cm300π cubic cm

    Volume is π r² h = π(25)(12) = 300π cubic cm. Using r rather than r² gives 60π, and doubling the radius before squaring gives 900π.Volume is π r^2 h = π(25)(12) = 300π cubic cm. Using r rather than r^2 gives 60π, and doubling the radius before squaring gives 900π.

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59. 59Right trianglesStandard

    A right triangle has legs of length 9 and 12. What is the length of its hypotenuse?A right triangle has legs of length 9 and 12. What is the length of its hypotenuse?

    • A2121
    • B1515
    • C1616
    • D1313
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    Answer: 1515

    9² + 12² = 81 + 144 = 225, and the square root of 225 is 15. Adding the legs gives 21, which is what the theorem exists to correct: the hypotenuse is always shorter than the two legs combined.9^2 + 12^2 = 81 + 144 = 225, and the square root of 225 is 15. Adding the legs gives 21, which is what the theorem exists to correct: the hypotenuse is always shorter than the two legs combined.

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60. 60Trigonometric ratiosStandard

    In right triangle ABC the right angle is at B, AB = 8 and BC = 15, so AC = 17. What is sin A?In right triangle ABC the right angle is at B, AB = 8 and BC = 15, so AC = 17. What is sin A?

    • A8178/17
    • B15815/8
    • C151715/17
    • D8158/15
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    Answer: 151715/17

    Sine is opposite over hypotenuse. The side opposite angle A is BC = 15 and the hypotenuse is AC = 17, so sin A = 1517. The choice 817 is cos A, taken from the side next to the angle rather than across from it.Sine is opposite over hypotenuse. The side opposite angle A is BC = 15 and the hypotenuse is AC = 17, so sin A = 15/17. The choice 8/17 is cos A, taken from the side next to the angle rather than across from it.

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61. 61Coordinate geometryStandard

    What is the distance between the points (-3, 2) and (5, 8)?What is the distance between the points (-3, 2) and (5, 8)?

    • A1414
    • Bthe square root of 68the square root of 68
    • C88
    • D1010
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    Answer: 1010

    The horizontal gap is 8 and the vertical gap is 6, so the distance is the square root of 64 + 36 = 100, which is 10. Adding the gaps gives 14, and subtracting the coordinates in the wrong order before squaring is what produces the square root of 68.The horizontal gap is 8 and the vertical gap is 6, so the distance is the square root of 64 + 36 = 100, which is 10. Adding the gaps gives 14, and subtracting the coordinates in the wrong order before squaring is what produces the square root of 68.

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62. 62Similar trianglesChallenge

    Triangle ABC is similar to triangle DEF, with AB = 6 and DE = 15. If the area of triangle ABC is 20, what is the area of triangle DEF?Triangle ABC is similar to triangle DEF, with AB = 6 and DE = 15. If the area of triangle ABC is 20, what is the area of triangle DEF?

    • A5050
    • B125125
    • C7575
    • D500500
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    Answer: 125125

    The sides scale by 156 = 52, so the areas scale by the square of that, 254. Then 20(254) = 125. Scaling the area by 52 instead gives 50, which is the error this question is built to catch.The sides scale by 15/6 = 5/2, so the areas scale by the square of that, 25/4. Then 20(25/4) = 125. Scaling the area by 5/2 instead gives 50, which is the error this question is built to catch.

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63. 63PolygonsStandard

    What is the measure of each interior angle of a regular decagon?What is the measure of each interior angle of a regular decagon?

    • A36°36°
    • B150°150°
    • C144°144°
    • D1,440°1,440°
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    Answer: 144°144°

    The interior angles of an n-sided polygon total (n - 2)(180), so for ten sides that is 8 × 180 = 1,440°. Divided among ten equal angles, each is 144°. The choice 36 is the exterior angle, and 1,440 is the total rather than one angle.The interior angles of an n-sided polygon total (n - 2)(180), so for ten sides that is 8 × 180 = 1,440°. Divided among ten equal angles, each is 144°. The choice 36 is the exterior angle, and 1,440 is the total rather than one angle.

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64. 64AreaChallenge

    A trapezium has parallel sides of 14 cm and 22 cm and a height of 9 cm. What is its area?A trapezium has parallel sides of 14 cm and 22 cm and a height of 9 cm. What is its area?

    • A324 square cm324 square cm
    • B126 square cm126 square cm
    • C198 square cm198 square cm
    • D162 square cm162 square cm
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    Answer: 162 square cm162 square cm

    The area is the average of the parallel sides times the height: 14 + 222 = 18, and 18 × 9 = 162 square cm. Forgetting to halve gives 324, and using one parallel side alone gives 126 or 198.The area is the average of the parallel sides times the height: (14 + 22)/2 = 18, and 18 × 9 = 162 square cm. Forgetting to halve gives 324, and using one parallel side alone gives 126 or 198.

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65. 65Special right trianglesStandard

    A 45-45-90 triangle has a hypotenuse of length 10 times the square root of 2. What is the length of each leg?A 45-45-90 triangle has a hypotenuse of length 10 times the square root of 2. What is the length of each leg?

    • A1010
    • B10 times the square root of 210 times the square root of 2
    • C5 times the square root of 25 times the square root of 2
    • D2020
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    Answer: 1010

    In a 45-45-90 triangle the hypotenuse is the leg times the square root of 2. Dividing the hypotenuse by the square root of 2 leaves 10. Multiplying by the square root of 2 instead goes the wrong way along the same relationship.In a 45-45-90 triangle the hypotenuse is the leg times the square root of 2. Dividing the hypotenuse by the square root of 2 leaves 10. Multiplying by the square root of 2 instead goes the wrong way along the same relationship.

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66. 66Angle measureFoundation

    Two angles are supplementary and one of them measures 47°. What does the other measure?Two angles are supplementary and one of them measures 47°. What does the other measure?

    • A43°43°
    • B133°133°
    • C53°53°
    • D313°313°
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    Answer: 133°133°

    Supplementary angles total 180°, so the other is 180 - 47 = 133. The choice 43 comes from using 90°, which is the rule for complementary angles.Supplementary angles total 180°, so the other is 180 - 47 = 133. The choice 43 comes from using 90°, which is the rule for complementary angles.

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67. 67Triangle anglesStandard

    In triangle ABC, angle A measures 3x degrees, angle B measures 2x degrees and angle C measures x degrees. What is the measure of angle A?In triangle ABC, angle A measures 3x degrees, angle B measures 2x degrees and angle C measures x degrees. What is the measure of angle A?

    • A30°30°
    • B60°60°
    • C90°90°
    • D120°120°
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    Answer: 90°90°

    The angles of a triangle total 180, so 3x + 2x + x = 6x = 180 and x = 30. Angle A is 3x = 90°. The choice 30 is x itself, which is angle C.The angles of a triangle total 180, so 3x + 2x + x = 6x = 180 and x = 30. Angle A is 3x = 90°. The choice 30 is x itself, which is angle C.

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68. 68AnglesStandard

    Two parallel lines are cut by a transversal, and one of the angles formed measures 118°. What is the measure of the same-side interior angle paired with it?Two parallel lines are cut by a transversal, and one of the angles formed measures 118°. What is the measure of the same-side interior angle paired with it?

    • A118°118°
    • B236°236°
    • C32°32°
    • D62°62°
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    Answer: 62°62°

    Same-side interior angles between parallel lines are supplementary, so the pair totals 180 and the other is 62°. The choice 118 is the measure of the angles that are equal to the given one rather than supplementary to it.Same-side interior angles between parallel lines are supplementary, so the pair totals 180 and the other is 62°. The choice 118 is the measure of the angles that are equal to the given one rather than supplementary to it.

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69. 69PolygonsChallenge

    Each interior angle of a regular polygon measures 156°. How many sides does it have?Each interior angle of a regular polygon measures 156°. How many sides does it have?

    • A1515
    • B1212
    • C1010
    • D1818
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    Answer: 1515

    Each exterior angle is 180 - 156 = 24°, and exterior angles of any polygon total 360, so there are 36024 = 15 sides. Checking forwards: (15 - 2)(180) = 2,340, and 2,34015 = 156.Each exterior angle is 180 - 156 = 24°, and exterior angles of any polygon total 360, so there are 360/24 = 15 sides. Checking forwards: (15 - 2)(180) = 2,340, and 2,340/15 = 156.

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70. 70Right trianglesStandard

    A ladder 13 m long leans against a vertical wall with its foot 5 m from the base of the wall. How far up the wall does the ladder reach?A ladder 13 m long leans against a vertical wall with its foot 5 m from the base of the wall. How far up the wall does the ladder reach?

    • A18 m18 m
    • B12 m12 m
    • C8 m8 m
    • Dthe square root of 194 mthe square root of 194 m
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    Answer: 12 m12 m

    The ladder is the hypotenuse, so the height is the square root of 13² - 5² = 169 - 25 = 144, which is 12 m. Adding the squares instead of subtracting them gives the square root of 194, the answer to a different arrangement.The ladder is the hypotenuse, so the height is the square root of 13^2 - 5^2 = 169 - 25 = 144, which is 12 m. Adding the squares instead of subtracting them gives the square root of 194, the answer to a different arrangement.

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71. 71CirclesChallenge

    A circle has its center at (3, -2) and passes through the point (7, 1). What is its radius?A circle has its center at (3, -2) and passes through the point (7, 1). What is its radius?

    • A2525
    • Bthe square root of 7the square root of 7
    • C55
    • D77
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    Answer: 55

    The radius is the distance from the center to the point: the gaps are 4 and 3, so the radius is the square root of 16 + 9 = 25, which is 5. The choice 25 stops one step early, at the square of the radius.The radius is the distance from the center to the point: the gaps are 4 and 3, so the radius is the square root of 16 + 9 = 25, which is 5. The choice 25 stops one step early, at the square of the radius.

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72. 72Coordinate geometryChallenge

    What is the midpoint of the segment joining (-5, 8) and (3, -2)?What is the midpoint of the segment joining (-5, 8) and (3, -2)?

    • A(-4, 5)(-4, 5)
    • B(-2, 6)(-2, 6)
    • C(4, -5)(4, -5)
    • D(-1, 3)(-1, 3)
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    Answer: (-1, 3)(-1, 3)

    A midpoint averages each coordinate separately: -5 + 32 = -1 and 8 + (-2)2 = 3. The choice (-2, 6) adds the coordinates without halving them, and (4, -5) halves the differences instead, which gives a displacement rather than a point.A midpoint averages each coordinate separately: (-5 + 3)/2 = -1 and (8 + (-2))/2 = 3. The choice (-2, 6) adds the coordinates without halving them, and (4, -5) halves the differences instead, which gives a displacement rather than a point.

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73. 73Special right trianglesStandard

    In a 30-60-90 triangle, the shorter leg has length 7. What is the length of the hypotenuse?In a 30-60-90 triangle, the shorter leg has length 7. What is the length of the hypotenuse?

    • A1414
    • B7 times the square root of 37 times the square root of 3
    • C7 times the square root of 27 times the square root of 2
    • D3.53.5
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    Answer: 1414

    In a 30-60-90 triangle the hypotenuse is twice the shorter leg, so it is 14. The square root of 3 belongs to the longer leg, and the square root of 2 to the 45-45-90 triangle.In a 30-60-90 triangle the hypotenuse is twice the shorter leg, so it is 14. The square root of 3 belongs to the longer leg, and the square root of 2 to the 45-45-90 triangle.

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74. 74Angle measureStandard

    Two angles are complementary, and one is twice the size of the other. What is the measure of the larger angle?Two angles are complementary, and one is twice the size of the other. What is the measure of the larger angle?

    • A30°30°
    • B60°60°
    • C120°120°
    • D45°45°
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    Answer: 60°60°

    Complementary angles total 90, so x + 2x = 90 gives x = 30 and the larger angle is 60. The choice 120 uses 180, which is the rule for supplementary angles.Complementary angles total 90, so x + 2x = 90 gives x = 30 and the larger angle is 60. The choice 120 uses 180, which is the rule for supplementary angles.

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75. 75Triangle anglesStandard

    An isosceles triangle has a vertex angle of 44°. What is the measure of each base angle?An isosceles triangle has a vertex angle of 44°. What is the measure of each base angle?

    • A44°44°
    • B136°136°
    • C68°68°
    • D46°46°
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    Answer: 68°68°

    The two base angles are equal and share what is left of 180: 180 - 442 = 68°. The choice 136 is the pair added together rather than one of them, and 46 comes from using 90 instead of 180.The two base angles are equal and share what is left of 180: (180 - 44)/2 = 68°. The choice 136 is the pair added together rather than one of them, and 46 comes from using 90 instead of 180.

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76. 76AnglesChallenge

    The four angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. What is the measure of the largest angle?The four angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. What is the measure of the largest angle?

    • A120°120°
    • B96°96°
    • C72°72°
    • D144°144°
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    Answer: 144°144°

    The angles of a quadrilateral total 360, and the ratio has 2 + 3 + 4 + 6 = 15 parts, so each part is 24°. The largest is 6 × 24 = 144. Using 180 for the total, as for a triangle, gives 72.The angles of a quadrilateral total 360, and the ratio has 2 + 3 + 4 + 6 = 15 parts, so each part is 24°. The largest is 6 × 24 = 144. Using 180 for the total, as for a triangle, gives 72.

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77. 77PolygonsStandard

    How many diagonals does a hexagon have?How many diagonals does a hexagon have?

    • A99
    • B66
    • C1212
    • D1515
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    Answer: 99

    Each of the 6 vertices joins to 3 others that are not itself or its two neighbors, giving 6 × 3 = 18 endpoints, and each diagonal has two, so there are 9. The choice 15 counts every pair of vertices, which includes the six sides.Each of the 6 vertices joins to 3 others that are not itself or its two neighbors, giving 6 × 3 = 18 endpoints, and each diagonal has two, so there are 9. The choice 15 counts every pair of vertices, which includes the six sides.

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78. 78CirclesChallenge

    A sector of a circle of radius 12 has a central angle of 60°. What is the area of the sector, in terms of π?A sector of a circle of radius 12 has a central angle of 60°. What is the area of the sector, in terms of π?

    • A12π12π
    • B24π24π
    • C48π48π
    • D144π144π
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    Answer: 24π24π

    A 60-degree sector is 60360 = one sixth of the circle. The whole circle has area π(12²) = 144π, so the sector has 144π / 6 = 24π. The choice 144π is the whole circle.A 60-degree sector is 60/360 = one sixth of the circle. The whole circle has area π(12^2) = 144π, so the sector has 144π / 6 = 24π. The choice 144π is the whole circle.

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79. 79VolumeStandard

    A cube has a volume of 343 cubic centimeters. Enter the length of one edge, in centimeters.A cube has a volume of 343 cubic centimeters. Enter the length of one edge, in centimeters.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 77

    The volume of a cube is the edge cubed, so the edge is the cube root of 343, which is 7 because 7 × 7 × 7 = 343.The volume of a cube is the edge cubed, so the edge is the cube root of 343, which is 7 because 7 × 7 × 7 = 343.

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80. 80AreaStandard

    A triangle has a base of 18 cm and a height of 11 cm. Enter its area in square centimeters.A triangle has a base of 18 cm and a height of 11 cm. Enter its area in square centimeters.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 9999

    The area of a triangle is half the base times the height: 18 × 112 = 1982 = 99 square cm. Forgetting to halve gives 198.The area of a triangle is half the base times the height: (18 × 11)/2 = 198/2 = 99 square cm. Forgetting to halve gives 198.

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81. 81AreaFoundation

    A rectangle has a perimeter of 34 cm and a width of 6 cm. What is its area?A rectangle has a perimeter of 34 cm and a width of 6 cm. What is its area?

    • A66 square cm66 square cm
    • B204 square cm204 square cm
    • C28 square cm28 square cm
    • D102 square cm102 square cm
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    Answer: 66 square cm66 square cm

    The perimeter is 2(length + width), so 2(length + 6) = 34 gives length = 11 cm. The area is then 11 × 6 = 66 square cm. Multiplying the perimeter by the width gives 204.The perimeter is 2(length + width), so 2(length + 6) = 34 gives length = 11 cm. The area is then 11 × 6 = 66 square cm. Multiplying the perimeter by the width gives 204.

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82. 82CirclesStandard

    A circle has an area of 49π. What is its circumference, in terms of π?A circle has an area of 49π. What is its circumference, in terms of π?

    • A7π7π
    • B14π14π
    • C49π49π
    • D98π98π
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    Answer: 14π14π

    From π r² = 49π comes r² = 49, so r = 7. The circumference is 2π r = 14π. Answering 7π uses the radius where the diameter belongs.From π r^2 = 49π comes r^2 = 49, so r = 7. The circumference is 2π r = 14π. Answering 7π uses the radius where the diameter belongs.

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83. 83VolumeStandard

    A rectangular prism measures 4 cm by 5 cm by 9 cm. What is its volume?A rectangular prism measures 4 cm by 5 cm by 9 cm. What is its volume?

    • A18 cubic cm18 cubic cm
    • B90 cubic cm90 cubic cm
    • C180 cubic cm180 cubic cm
    • D360 cubic cm360 cubic cm
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    Answer: 180 cubic cm180 cubic cm

    Volume is the product of the three dimensions: 4 × 5 × 9 = 180 cubic cm. The choice 18 adds them instead, and 90 multiplies only two.Volume is the product of the three dimensions: 4 × 5 × 9 = 180 cubic cm. The choice 18 adds them instead, and 90 multiplies only two.

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84. 84Trigonometric ratiosChallenge

    In a right triangle, the tangent of angle theta is 34 and the side opposite theta has length 9. What is the length of the side adjacent to theta?In a right triangle, the tangent of angle theta is 3/4 and the side opposite theta has length 9. What is the length of the side adjacent to theta?

    • A6.756.75
    • B1515
    • C33
    • D1212
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    Answer: 1212

    Tangent is opposite over adjacent, so 9 / adjacent = 34 and adjacent = 9(43) = 12. Multiplying by 34 instead of dividing gives 6.75, and 15 is the hypotenuse of the 9-12-15 triangle this turns out to be.Tangent is opposite over adjacent, so 9 / adjacent = 3/4 and adjacent = 9(4/3) = 12. Multiplying by 3/4 instead of dividing gives 6.75, and 15 is the hypotenuse of the 9-12-15 triangle this turns out to be.

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85. 85Similar trianglesStandard

    Two similar triangles have corresponding sides of 8 cm and 20 cm. The smaller triangle has a perimeter of 24 cm. What is the perimeter of the larger?Two similar triangles have corresponding sides of 8 cm and 20 cm. The smaller triangle has a perimeter of 24 cm. What is the perimeter of the larger?

    • A150 cm150 cm
    • B9.6 cm9.6 cm
    • C60 cm60 cm
    • D48 cm48 cm
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    Answer: 60 cm60 cm

    Perimeter is a length, so it scales by the same factor as the sides: 208 = 2.5, and 24 × 2.5 = 60 cm. Squaring the factor, which is right for areas, gives 150.Perimeter is a length, so it scales by the same factor as the sides: 20/8 = 2.5, and 24 × 2.5 = 60 cm. Squaring the factor, which is right for areas, gives 150.

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86. 86Coordinate geometryChallenge

    A line passes through (2, 3) and is perpendicular to the line y = -(14)x + 7. What is its slope?A line passes through (2, 3) and is perpendicular to the line y = -(1/4)x + 7. What is its slope?

    • A-14-1/4
    • B-4-4
    • C141/4
    • D44
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    Answer: 44

    Perpendicular slopes are negative reciprocals: the negative reciprocal of -14 is 4. Taking only the reciprocal gives -4 and only the negative gives 14; the point (2, 3) fixes the intercept, not the slope.Perpendicular slopes are negative reciprocals: the negative reciprocal of -1/4 is 4. Taking only the reciprocal gives -4 and only the negative gives 1/4; the point (2, 3) fixes the intercept, not the slope.

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87. 87AnglesStandard

    Two angles of a triangle measure 37° and 58°. Enter the measure of the third angle in degrees.Two angles of a triangle measure 37° and 58°. Enter the measure of the third angle in degrees.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 8585

    The three angles total 180, so the third is 180 - 37 - 58 = 85°.The three angles total 180, so the third is 180 - 37 - 58 = 85°.

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88. 88Right trianglesStandard

    A right triangle has a hypotenuse of 25 and one leg of 7. Enter the length of the other leg.A right triangle has a hypotenuse of 25 and one leg of 7. Enter the length of the other leg.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 2424

    The other leg is the square root of 25² - 7² = 625 - 49 = 576, which is 24. Because the hypotenuse is given, the two squares are subtracted rather than added.The other leg is the square root of 25^2 - 7^2 = 625 - 49 = 576, which is 24. Because the hypotenuse is given, the two squares are subtracted rather than added.

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89. 89CirclesFoundation

    A circle has a diameter of 18 cm. What is its area, in terms of π?A circle has a diameter of 18 cm. What is its area, in terms of π?

    • A81π square cm81π square cm
    • B324π square cm324π square cm
    • C36π square cm36π square cm
    • D18π square cm18π square cm
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    Answer: 81π square cm81π square cm

    The radius is half the diameter, so 9 cm, and the area is π(9²) = 81π square cm. Using the diameter in place of the radius gives 324π, four times too large.The radius is half the diameter, so 9 cm, and the area is π(9^2) = 81π square cm. Using the diameter in place of the radius gives 324π, four times too large.

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90. 90VolumeStandard

    A cone has a radius of 3 cm and a height of 7 cm. What is its volume, in terms of π?A cone has a radius of 3 cm and a height of 7 cm. What is its volume, in terms of π?

    • A63π cubic cm63π cubic cm
    • B21π cubic cm21π cubic cm
    • C7π cubic cm7π cubic cm
    • D9π cubic cm9π cubic cm
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    Answer: 21π cubic cm21π cubic cm

    The volume of a cone is one third of π r² h: (13)(π)(9)(7) = 21π cubic cm. Omitting the one third gives 63π, which is the cylinder the cone sits inside.The volume of a cone is one third of π r^2 h: (1/3)(π)(9)(7) = 21π cubic cm. Omitting the one third gives 63π, which is the cylinder the cone sits inside.

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91. 91Special right trianglesChallenge

    In a 30-60-90 triangle the shorter leg is 5 times the square root of 3. What is the length of the longer leg?In a 30-60-90 triangle the shorter leg is 5 times the square root of 3. What is the length of the longer leg?

    • A55
    • B10 times the square root of 310 times the square root of 3
    • C1515
    • D5 times the square root of 35 times the square root of 3
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    Answer: 1515

    The longer leg is the shorter leg times the square root of 3, so it is 5 times the square root of 3, times the square root of 3, which is 5 × 3 = 15. The two roots multiply to 3 rather than staying under the radical.The longer leg is the shorter leg times the square root of 3, so it is 5 times the square root of 3, times the square root of 3, which is 5 × 3 = 15. The two roots multiply to 3 rather than staying under the radical.

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92. 92Coordinate geometryStandard

    What is the slope of the line through (-2, -5) and (4, 7)?What is the slope of the line through (-2, -5) and (4, 7)?

    • A121/2
    • B-2-2
    • C66
    • D22
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    Answer: 22

    The change in y is 7 - (-5) = 12 and the change in x is 4 - (-2) = 6, so the slope is 126 = 2. Both subtractions involve a double negative, which is where the sign errors come from.The change in y is 7 - (-5) = 12 and the change in x is 4 - (-2) = 6, so the slope is 12/6 = 2. Both subtractions involve a double negative, which is where the sign errors come from.

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93. 93Triangle anglesStandard

    One exterior angle of a triangle measures 115°. What is the sum of the two interior angles not adjacent to it?One exterior angle of a triangle measures 115°. What is the sum of the two interior angles not adjacent to it?

    • A115°115°
    • B65°65°
    • C245°245°
    • D57.5°57.5°
    Show the answer and explanationHide the explanation

    Answer: 115°115°

    An exterior angle equals the sum of the two opposite interior angles, so the sum is 115°. The choice 65 is the adjacent interior angle, which is the one this rule does not involve.An exterior angle equals the sum of the two opposite interior angles, so the sum is 115°. The choice 65 is the adjacent interior angle, which is the one this rule does not involve.

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94. 94AreaStandard

    A parallelogram has a base of 14 cm and a perpendicular height of 9 cm. What is its area?A parallelogram has a base of 14 cm and a perpendicular height of 9 cm. What is its area?

    • A63 square cm63 square cm
    • B126 square cm126 square cm
    • C23 square cm23 square cm
    • D46 square cm46 square cm
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    Answer: 126 square cm126 square cm

    The area of a parallelogram is base times perpendicular height: 14 × 9 = 126 square cm. Halving it, as for a triangle, gives 63.The area of a parallelogram is base times perpendicular height: 14 × 9 = 126 square cm. Halving it, as for a triangle, gives 63.

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95. 95AnglesStandard

    A regular pentagon has how many degrees in each of its exterior angles? Enter the number.A regular pentagon has how many degrees in each of its exterior angles? Enter the number.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 7272

    The exterior angles of any polygon total 360°, and a regular pentagon has five equal ones: 3605 = 72°.The exterior angles of any polygon total 360°, and a regular pentagon has five equal ones: 360/5 = 72°.

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96. 96Right trianglesStandard

    A right triangle has legs of 20 and 21. Enter the length of its hypotenuse.A right triangle has legs of 20 and 21. Enter the length of its hypotenuse.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 2929

    20² + 21² = 400 + 441 = 841, and the square root of 841 is 29. The 20-21-29 triangle is one of the less familiar right triangles with whole-number sides.20^2 + 21^2 = 400 + 441 = 841, and the square root of 841 is 29. The 20-21-29 triangle is one of the less familiar right triangles with whole-number sides.

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97. 97CirclesStandard

    A circle has a circumference of 44π. What is its radius?A circle has a circumference of 44π. What is its radius?

    • A4444
    • B1111
    • C2222
    • D8888
    Show the answer and explanationHide the explanation

    Answer: 2222

    Circumference is 2π r, so 2r = 44 and r = 22. Halving twice gives 11, which is the radius of a circle with half this circumference.Circumference is 2π r, so 2r = 44 and r = 22. Halving twice gives 11, which is the radius of a circle with half this circumference.

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98. 98Right trianglesFoundation

    A right triangle has legs of 7 and 24. What is its hypotenuse?A right triangle has legs of 7 and 24. What is its hypotenuse?

    • A3131
    • B2323
    • C2626
    • D2525
    Show the answer and explanationHide the explanation

    Answer: 2525

    7² + 24² = 49 + 576 = 625, and the square root of 625 is 25. Adding the legs gives 31; the hypotenuse is always less than that.7^2 + 24^2 = 49 + 576 = 625, and the square root of 625 is 25. Adding the legs gives 31; the hypotenuse is always less than that.

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99. 99PolygonsStandard

    What is the measure of each interior angle of a regular octagon?What is the measure of each interior angle of a regular octagon?

    • A135°135°
    • B45°45°
    • C120°120°
    • D1,080°1,080°
    Show the answer and explanationHide the explanation

    Answer: 135°135°

    The interior angles total (8 - 2)(180) = 1,080°, and dividing among eight equal angles gives 135. The choice 45 is the exterior angle and 1,080 is the total.The interior angles total (8 - 2)(180) = 1,080°, and dividing among eight equal angles gives 135. The choice 45 is the exterior angle and 1,080 is the total.

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100. 100Coordinate geometryStandard

     What is the midpoint of the segment joining (3, -7) and (11, 1)?What is the midpoint of the segment joining (3, -7) and (11, 1)?

     • A(7, -3)(7, -3)
     • B(8, -6)(8, -6)
     • C(14, -6)(14, -6)
     • D(4, -4)(4, -4)
     Show the answer and explanationHide the explanation

     Answer: (7, -3)(7, -3)

     Average each coordinate: 3 + 112 = 7 and -7 + 12 = -3. Adding without halving gives (14, -6), and halving the differences instead gives a displacement rather than a point.Average each coordinate: (3 + 11)/2 = 7 and (-7 + 1)/2 = -3. Adding without halving gives (14, -6), and halving the differences instead gives a displacement rather than a point.

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101. 101AreaChallenge

     A closed cylinder has a radius of 5 and a height of 9. What is its total surface area, in terms of π?A closed cylinder has a radius of 5 and a height of 9. What is its total surface area, in terms of π?

     • A90π90π
     • B140π140π
     • C115π115π
     • D225π225π
     Show the answer and explanationHide the explanation

     Answer: 140π140π

     The curved surface is 2π r h = 90π, and the two circular ends add 2π r² = 50π, giving 140π. Leaving out one end gives 115π and leaving out both gives 90π.The curved surface is 2π r h = 90π, and the two circular ends add 2π r^2 = 50π, giving 140π. Leaving out one end gives 115π and leaving out both gives 90π.

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102. 102Trigonometric ratiosStandard

     In a right triangle, the side adjacent to angle theta is 12 and the hypotenuse is 13. What is cos theta?In a right triangle, the side adjacent to angle theta is 12 and the hypotenuse is 13. What is cos theta?

     • A5135/13
     • B5125/12
     • C121312/13
     • D131213/12
     Show the answer and explanationHide the explanation

     Answer: 121312/13

     Cosine is adjacent over hypotenuse, so 1213. The remaining side is 5, making 513 the sine and 512 the tangent — all three are available, and the question asks for one.Cosine is adjacent over hypotenuse, so 12/13. The remaining side is 5, making 5/13 the sine and 5/12 the tangent — all three are available, and the question asks for one.

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103. 103CirclesChallenge

     A sector of a circle of radius 8 has a central angle of 90°. What is its area, in terms of π?A sector of a circle of radius 8 has a central angle of 90°. What is its area, in terms of π?

     • A64π64π
     • B32π32π
     • C8π8π
     • D16π16π
     Show the answer and explanationHide the explanation

     Answer: 16π16π

     A 90-degree sector is a quarter of the circle. The whole circle has area 64π, so the sector has 16π. The choice 64π is the whole circle and 32π is half of it.A 90-degree sector is a quarter of the circle. The whole circle has area 64π, so the sector has 16π. The choice 64π is the whole circle and 32π is half of it.

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104. 104Similar trianglesChallenge

     Two similar figures have corresponding sides in the ratio 3 : 7. What is the ratio of their areas?Two similar figures have corresponding sides in the ratio 3 : 7. What is the ratio of their areas?

     • A9 : 499 : 49
     • B3 : 73 : 7
     • C6 : 146 : 14
     • D27 : 34327 : 343
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     Answer: 9 : 499 : 49

     Areas scale by the square of the length ratio, so 3² : 7² = 9 : 49. The last choice cubes the ratio, which is right for volumes and not for areas.Areas scale by the square of the length ratio, so 3^2 : 7^2 = 9 : 49. The last choice cubes the ratio, which is right for volumes and not for areas.

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105. 105CirclesFoundation

     A circle has an area of 121π. What is its radius?A circle has an area of 121π. What is its radius?

     • A60.560.5
     • B1111
     • C121121
     • D2222
     Show the answer and explanationHide the explanation

     Answer: 1111

     From π r² = 121π comes r² = 121, so r = 11. The choice 121 stops before taking the square root, and 22 is the diameter.From π r^2 = 121π comes r^2 = 121, so r = 11. The choice 121 stops before taking the square root, and 22 is the diameter.

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106. 106Right trianglesFoundation

     A right triangle has legs of 15 and 20. What is its hypotenuse?A right triangle has legs of 15 and 20. What is its hypotenuse?

     • A3535
     • B3030
     • C2525
     • D2727
     Show the answer and explanationHide the explanation

     Answer: 2525

     15² + 20² = 225 + 400 = 625, and the square root of 625 is 25. This is the 3-4-5 triangle scaled by 5, which is worth recognizing on sight.15^2 + 20^2 = 225 + 400 = 625, and the square root of 625 is 25. This is the 3-4-5 triangle scaled by 5, which is worth recognizing on sight.

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107. 107PolygonsStandard

     What is the measure of each interior angle of a regular nonagon?What is the measure of each interior angle of a regular nonagon?

     • A40°40°
     • B1,260°1,260°
     • C120°120°
     • D140°140°
     Show the answer and explanationHide the explanation

     Answer: 140°140°

     The interior angles total (9 - 2)(180) = 1,260°, and dividing among nine equal angles gives 140. The choice 40 is the exterior angle and 1,260 is the total.The interior angles total (9 - 2)(180) = 1,260°, and dividing among nine equal angles gives 140. The choice 40 is the exterior angle and 1,260 is the total.

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108. 108Coordinate geometryStandard

     What is the midpoint of the segment joining (-6, 3) and (2, 11)?What is the midpoint of the segment joining (-6, 3) and (2, 11)?

     • A(-2, 7)(-2, 7)
     • B(-4, 14)(-4, 14)
     • C(4, -4)(4, -4)
     • D(-8, -8)(-8, -8)
     Show the answer and explanationHide the explanation

     Answer: (-2, 7)(-2, 7)

     Average each coordinate: -6 + 22 = -2 and 3 + 112 = 7. Adding without halving gives (-4, 14).Average each coordinate: (-6 + 2)/2 = -2 and (3 + 11)/2 = 7. Adding without halving gives (-4, 14).

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109. 109VolumeStandard

     A cylinder has a radius of 6 and a height of 4. What is its volume, in terms of π?A cylinder has a radius of 6 and a height of 4. What is its volume, in terms of π?

     • A48π48π
     • B144π144π
     • C96π96π
     • D576π576π
     Show the answer and explanationHide the explanation

     Answer: 144π144π

     Volume is π r² h = π(36)(4) = 144π. Using r rather than r² gives 24π, and doubling the radius before squaring gives 576π.Volume is π r^2 h = π(36)(4) = 144π. Using r rather than r^2 gives 24π, and doubling the radius before squaring gives 576π.

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110. 110Trigonometric ratiosStandard

     In a right triangle, the side opposite angle theta is 9 and the hypotenuse is 15. What is sin theta?In a right triangle, the side opposite angle theta is 9 and the hypotenuse is 15. What is sin theta?

     • A454/5
     • B343/4
     • C353/5
     • D535/3
     Show the answer and explanationHide the explanation

     Answer: 353/5

     Sine is opposite over hypotenuse: 915 = 35. The remaining side is 12, making 45 the cosine and 34 the tangent — all three are available and the question asks for one.Sine is opposite over hypotenuse: 9/15 = 3/5. The remaining side is 12, making 4/5 the cosine and 3/4 the tangent — all three are available and the question asks for one.

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111. 111CirclesChallenge

     A sector of a circle of radius 9 has a central angle of 120°. Enter the area of the sector as a multiple of π.A sector of a circle of radius 9 has a central angle of 120°. Enter the area of the sector as a multiple of π.

     Student-produced response: write your own answer rather than choosing one.

     Show the answer and explanationHide the explanation

     Answer: 2727

     A 120-degree sector is one third of the circle. The whole circle has area 81π, so the sector has 27π.A 120-degree sector is one third of the circle. The whole circle has area 81π, so the sector has 27π.

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112. 112Similar trianglesChallenge

     Two similar solids have corresponding edges in the ratio 5 : 8. What is the ratio of their volumes?Two similar solids have corresponding edges in the ratio 5 : 8. What is the ratio of their volumes?

     • A5 : 85 : 8
     • B25 : 6425 : 64
     • C10 : 1610 : 16
     • D125 : 512125 : 512
     Show the answer and explanationHide the explanation

     Answer: 125 : 512125 : 512

     Volumes scale by the cube of the length ratio: 5³ : 8³ = 125 : 512. Squaring instead gives 25 : 64, which is the ratio of their surface areas.Volumes scale by the cube of the length ratio: 5^3 : 8^3 = 125 : 512. Squaring instead gives 25 : 64, which is the ratio of their surface areas.

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