Geometry & Trigonometry practice questions

Geometry & Trigonometry questions

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1. 1Triangle anglesFoundation

   Two angles of a triangle measure 48° and 67°. What is the measure of the third angle?Two angles of a triangle measure 48° and 67°. What is the measure of the third angle?

   • A6060
   • B5555
   • C6565
   • D115115
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   Answer: 6565

   The angles of a triangle total 180°. 180 - 48 - 67 = 65.The angles of a triangle total 180°. 180 - 48 - 67 = 65.

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2. 2AreaFoundation

   What is the area of a rectangle with length 8 and width 5?What is the area of a rectangle with length 8 and width 5?

   • A4040
   • B2626
   • C8080
   • D1313
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   Answer: 4040

   Area of a rectangle is length times width: 8 × 5 = 40. The perimeter of the same rectangle is 2(8) + 2(5) = 26, so read which one the question asked for.Area of a rectangle is length times width: 8 × 5 = 40. The perimeter of the same rectangle is 2(8) + 2(5) = 26, so read which one the question asked for.

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3. 3CirclesStandard

   A circle has radius 4. What is its circumference in terms of π?A circle has radius 4. What is its circumference in terms of π?

   • A32π32π
   • B4π4π
   • C16π16π
   • D8π8π
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   Answer: 8π8π

   Circumference is 2π r. With r = 4, the circumference is 8π.Circumference is 2π r. With r = 4, the circumference is 8π.

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4. 4Right trianglesStandard

   A right triangle has legs of length 5 and 12. What is the length of its hypotenuse?A right triangle has legs of length 5 and 12. What is the length of its hypotenuse?

   • A2525
   • B1717
   • C1515
   • D1313
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   Answer: 1313

   Use the Pythagorean theorem: 5² + 12² = 25 + 144 = 169. The square root of 169 is 13.Use the Pythagorean theorem: 5^2 + 12^2 = 25 + 144 = 169. The square root of 169 is 13.

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5. 5AreaStandard

   A trapezoid has bases of length 6 and 10 and height 4. What is its area?A trapezoid has bases of length 6 and 10 and height 4. What is its area?

   • A6464
   • B1616
   • C2424
   • D3232
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   Answer: 3232

   Area = one-half times the sum of the bases times the height: 6 + 102 = 8, and 8 × 4 = 32.Area = one-half times the sum of the bases times the height: (6 + 10)/2 = 8, and 8 × 4 = 32.

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6. 6Similar trianglesStandard

   Two similar triangles have corresponding side lengths in the ratio 3:5. If the longer corresponding side is 15, what is the shorter side?Two similar triangles have corresponding side lengths in the ratio 3:5. If the longer corresponding side is 15, what is the shorter side?

   • A99
   • B1212
   • C1010
   • D66
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   Answer: 99

   The scale factor from 5 to 15 is 3. Apply it to 3: 3 × 3 = 9.The scale factor from 5 to 15 is 3. Apply it to 3: 3 × 3 = 9.

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7. 7VolumeFoundation

   A cube has side length 3. Enter its volume.A cube has side length 3. Enter its volume.

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

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   Answer: 2727

   Volume of a cube is the side cubed: 3³ = 27. Its surface area is 6(3²) = 54, which answers a different question and is the commonest wrong choice here.Volume of a cube is the side cubed: 3^3 = 27. Its surface area is 6(3^2) = 54, which answers a different question and is the commonest wrong choice here.

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8. 8Coordinate geometryFoundation

   What is the distance between (-1, 2) and (3, 2) on a coordinate plane?What is the distance between (-1, 2) and (3, 2) on a coordinate plane?

   • A22
   • B55
   • C44
   • D88
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   Answer: 44

   The points have the same y-coordinate, so the horizontal distance is 3 - (-1) = 4.The points have the same y-coordinate, so the horizontal distance is 3 - (-1) = 4.

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9. 9AreaFoundation

   A square has perimeter 36. What is its area?A square has perimeter 36. What is its area?

   • A3636
   • B1818
   • C7272
   • D8181
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   Answer: 8181

   Work back to the side first: a perimeter of 36 means each side is 364 = 9, so the area is 9² = 81. Two steps, and the first is the one that gets skipped.Work back to the side first: a perimeter of 36 means each side is 36/4 = 9, so the area is 9^2 = 81. Two steps, and the first is the one that gets skipped.

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10. 10AreaFoundation

    A triangle has base 10 and height 7. What is its area?A triangle has base 10 and height 7. What is its area?

    • A17.517.5
    • B3535
    • C140140
    • D7070
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    Answer: 3535

    Triangle area is one-half times base times height: 10 × 7 = 70, and 702 = 35.Triangle area is one-half times base times height: 10 × 7 = 70, and 70/2 = 35.

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11. 11VolumeStandard

    A cylinder has radius 2 and height 5. What is its volume in terms of π?A cylinder has radius 2 and height 5. What is its volume in terms of π?

    • A25π25π
    • B10π10π
    • C20π20π
    • D40π40π
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    Answer: 20π20π

    Cylinder volume is π r² h: π(2²)(5) = 20π. Square the radius before multiplying by the height — doubling instead of squaring gives 20π by luck here, but not generally.Cylinder volume is π r^2 h: π(2^2)(5) = 20π. Square the radius before multiplying by the height — doubling instead of squaring gives 20π by luck here, but not generally.

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12. 12CirclesFoundation

    A circle has diameter 10. What is its area in terms of π?A circle has diameter 10. What is its area in terms of π?

    • A5π5π
    • B10π10π
    • C25π25π
    • D100π100π
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    Answer: 25π25π

    The radius is half the diameter, so r = 5. Area is π r² = 25π.The radius is half the diameter, so r = 5. Area is π r^2 = 25π.

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13. 13Right trianglesFoundation

    A right triangle has one leg of length 6 and a hypotenuse of length 10. Enter the length of the other leg.A right triangle has one leg of length 6 and a hypotenuse of length 10. Enter the length of the other leg.

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

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    Answer: 88

    By the Pythagorean theorem, 6² + b² = 10², so b² = 100 - 36 = 64 and b = 8.By the Pythagorean theorem, 6^2 + b^2 = 10^2, so b^2 = 100 - 36 = 64 and b = 8.

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14. 14Trigonometric ratiosStandard

    In a right triangle, the side opposite angle A has length 3 and the hypotenuse has length 5. What is sin A?In a right triangle, the side opposite angle A has length 3 and the hypotenuse has length 5. What is sin A?

    • A353/5
    • B454/5
    • C343/4
    • D535/3
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    Answer: 353/5

    Sine is the ratio of the opposite side to the hypotenuse, which is 35.Sine is the ratio of the opposite side to the hypotenuse, which is 3/5.

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15. 15AnglesFoundation

    Two parallel lines are cut by a transversal, and one angle measures 65°. What is the measure of its corresponding angle?Two parallel lines are cut by a transversal, and one angle measures 65°. What is the measure of its corresponding angle?

    • A130°130°
    • B115°115°
    • C65°65°
    • D25°25°
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    Answer: 65°65°

    Corresponding angles formed by a transversal crossing parallel lines are congruent, so the angle also measures 65°.Corresponding angles formed by a transversal crossing parallel lines are congruent, so the angle also measures 65°.

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16. 16VolumeChallenge

    A sphere has radius 3. What is its volume in terms of π?A sphere has radius 3. What is its volume in terms of π?

    • A36π36π
    • B12π12π
    • C108π108π
    • D27π27π
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    Answer: 36π36π

    The volume of a sphere is (43)π r³. With r = 3, r³ = 27, so the volume is (43)(27)π = 36π.The volume of a sphere is (4/3)π r^3. With r = 3, r^3 = 27, so the volume is (4/3)(27)π = 36π.

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17. 17Coordinate geometryStandard

    What is the midpoint of the segment joining (2, -1) and (8, 5)?What is the midpoint of the segment joining (2, -1) and (8, 5)?

    • A(3, 3)(3, 3)
    • B(5, 2)(5, 2)
    • C(6, 6)(6, 6)
    • D(10, 4)(10, 4)
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    Answer: (5, 2)(5, 2)

    Average each coordinate: x is 2 + 82 = 5 and y is -1 + 52 = 2.Average each coordinate: x is (2 + 8)/2 = 5 and y is (-1 + 5)/2 = 2.

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18. 18Similar trianglesStandard

    Triangle ABC is similar to triangle DEF, with AB = 4, DE = 10, and BC = 6. Enter the length of EF.Triangle ABC is similar to triangle DEF, with AB = 4, DE = 10, and BC = 6. Enter the length of EF.

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

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    Answer: 1515

    The scale factor from ABC to DEF is 104 = 2.5. Applying it to BC gives EF = 6 × 2.5 = 15.The scale factor from ABC to DEF is 10/4 = 2.5. Applying it to BC gives EF = 6 × 2.5 = 15.

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19. 19CirclesChallenge

    If a circle's radius is doubled, by what factor does its area increase?If a circle's radius is doubled, by what factor does its area increase?

    • A44
    • B88
    • C1616
    • D22
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    Answer: 44

    Area is π r². Replacing r with 2r gives π(2r)² = 4π r², so the area is 4 times as large.Area is π r^2. Replacing r with 2r gives π(2r)^2 = 4π r^2, so the area is 4 times as large.

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20. 20Coordinate geometryFoundation

    Enter the distance between the points (0, 0) and (6, 8).Enter the distance between the points (0, 0) and (6, 8).

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

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    Answer: 1010

    The distance is the square root of 6² + 8² = 36 + 64 = 100, which is 10.The distance is the square root of 6^2 + 8^2 = 36 + 64 = 100, which is 10.

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21. 21PolygonsStandard

    What is the sum of the interior angles of a pentagon?What is the sum of the interior angles of a pentagon?

    • A720°720°
    • B360°360°
    • C900°900°
    • D540°540°
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    Answer: 540°540°

    The sum of the interior angles of a polygon with n sides is (n - 2)(180) degrees. For n = 5, that is 3 × 180 = 540°.The sum of the interior angles of a polygon with n sides is (n - 2)(180) degrees. For n = 5, that is 3 × 180 = 540°.

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22. 22Trigonometric ratiosStandard

    In a right triangle, the leg opposite angle B has length 7 and the leg adjacent to angle B has length 24. What is tan B?In a right triangle, the leg opposite angle B has length 7 and the leg adjacent to angle B has length 24. What is tan B?

    • A7247/24
    • B24724/7
    • C7257/25
    • D242524/25
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    Answer: 7247/24

    Tangent is the ratio of the opposite leg to the adjacent leg, which is 724. The hypotenuse is not used.Tangent is the ratio of the opposite leg to the adjacent leg, which is 7/24. The hypotenuse is not used.

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23. 23VolumeStandard

    A cone has radius 3 and height 4. What is its volume in terms of π?A cone has radius 3 and height 4. What is its volume in terms of π?

    • A9π9π
    • B16π16π
    • C12π12π
    • D36π36π
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    Answer: 12π12π

    The volume of a cone is (13)π r² h. With r = 3 and h = 4, that is (13)(9)(4)π = 12π, which is one third of the cylinder with the same base and height.The volume of a cone is (1/3)π r^2 h. With r = 3 and h = 4, that is (1/3)(9)(4)π = 12π, which is one third of the cylinder with the same base and height.

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24. 24Special right trianglesStandard

    In a 30-60-90 triangle, the shortest side has length 5. What is the length of the hypotenuse?In a 30-60-90 triangle, the shortest side has length 5. What is the length of the hypotenuse?

    • A1010
    • B5 √35 √(3)
    • C1515
    • D5 √25 √(2)
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    Answer: 1010

    The sides of a 30-60-90 triangle are in the ratio x, x √3, 2x, with the hypotenuse twice the shortest side. With x = 5, the hypotenuse is 10.The sides of a 30-60-90 triangle are in the ratio x, x √(3), 2x, with the hypotenuse twice the shortest side. With x = 5, the hypotenuse is 10.

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25. 25Trigonometric ratiosChallenge

    In a right triangle, angle A and angle B are the two acute angles. If sin A = 0.6, what is cos B?In a right triangle, angle A and angle B are the two acute angles. If sin A = 0.6, what is cos B?

    • A1.61.6
    • B0.60.6
    • C0.40.4
    • D0.80.8
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    Answer: 0.60.6

    The two acute angles in a right triangle are complementary, and sin(x) = cos(90 - x). Because B = 90 - A, cos B equals sin A, which is 0.6.The two acute angles in a right triangle are complementary, and sin(x) = cos(90 - x). Because B = 90 - A, cos B equals sin A, which is 0.6.

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26. 26Angle measureFoundation

    An angle measures 90°. What is its measure in radians?An angle measures 90°. What is its measure in radians?

    • Aπ/2π/2
    • Bπ/4π/4
    • C2π2π
    • Dππ
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    Answer: π/2π/2

    Because 180° equals π radians, multiply by π/180: 90(π/180) = π/2.Because 180° equals π radians, multiply by π/180: 90(π/180) = π/2.

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27. 27CirclesChallenge

    A circle has area 36π. Enter its circumference divided by π.A circle has area 36π. Enter its circumference divided by π.

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

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    Answer: 1212

    From π r² = 36π, r² = 36 and r = 6. The circumference is 2π r = 12π, so dividing by π gives 12.From π r^2 = 36π, r^2 = 36 and r = 6. The circumference is 2π r = 12π, so dividing by π gives 12.

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28. 28Triangle anglesFoundation

    In an isosceles triangle, the two equal angles each measure 52°. What is the measure of the third angle?In an isosceles triangle, the two equal angles each measure 52°. What is the measure of the third angle?

    • A76°76°
    • B104°104°
    • C128°128°
    • D52°52°
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    Answer: 76°76°

    The three angles total 180°. Two of them account for 104°, so the third is 180 - 104 = 76°.The three angles total 180°. Two of them account for 104°, so the third is 180 - 104 = 76°.

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29. 29VolumeStandard

    A rectangular tank measures 4 by 3 by 2 meters. If its height is doubled and the other dimensions are unchanged, what happens to its volume?A rectangular tank measures 4 by 3 by 2 meters. If its height is doubled and the other dimensions are unchanged, what happens to its volume?

    • AIt increases by 8 times.It increases by 8 times.
    • BIt doubles.It doubles.
    • CIt quadruples.It quadruples.
    • DIt is unchanged.It is unchanged.
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    Answer: It doubles.It doubles.

    Volume is length times width times height. Scaling only one dimension by 2 multiplies the volume by 2. Doubling all three would multiply it by 8.Volume is length times width times height. Scaling only one dimension by 2 multiplies the volume by 2. Doubling all three would multiply it by 8.

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30. 30Coordinate geometryStandard

    A circle has center (3, -1) and passes through (3, 4). What is its radius?A circle has center (3, -1) and passes through (3, 4). What is its radius?

    • A√10√(10)
    • B55
    • C33
    • D44
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    Answer: 55

    The two points share an x-coordinate, so the distance between them is the difference in y: 4 - (-1) = 5. No distance formula is needed.The two points share an x-coordinate, so the distance between them is the difference in y: 4 - (-1) = 5. No distance formula is needed.

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31. 31Similar trianglesChallenge

    A 1.8-meter person casts a 2.4-meter shadow at the same moment a tree casts a 20-meter shadow. Enter the height of the tree in meters.A 1.8-meter person casts a 2.4-meter shadow at the same moment a tree casts a 20-meter shadow. Enter the height of the tree in meters.

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

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    Answer: 1515

    The person and the tree form similar right triangles with the sun's rays, so height over shadow is constant: 1.82.4 = 0.75. The tree's height is 0.75 × 20 = 15 meters.The person and the tree form similar right triangles with the sun's rays, so height over shadow is constant: 1.8/2.4 = 0.75. The tree's height is 0.75 × 20 = 15 meters.

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32. 32Trigonometric ratiosStandard

    In a right triangle, the legs measure 8 and 15. What is the cosine of the angle adjacent to the leg of length 8?In a right triangle, the legs measure 8 and 15. What is the cosine of the angle adjacent to the leg of length 8?

    • A15815/8
    • B8178/17
    • C8158/15
    • D151715/17
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    Answer: 8178/17

    The hypotenuse is √64 + 225 = √289 = 17. Cosine is the adjacent leg over the hypotenuse, and the leg adjacent to that angle is 8, giving 817.The hypotenuse is √(64 + 225) = √(289) = 17. Cosine is the adjacent leg over the hypotenuse, and the leg adjacent to that angle is 8, giving 8/17.

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33. 33CirclesStandard

    A circle has center (2, -3) and radius 5. Which equation describes it?A circle has center (2, -3) and radius 5. Which equation describes it?

    • A(x - 2)² + (y + 3)² = 25(x - 2)^2 + (y + 3)^2 = 25
    • B(x - 2)² + (y + 3)² = 5(x - 2)^2 + (y + 3)^2 = 5
    • C(x + 2)² + (y - 3)² = 25(x + 2)^2 + (y - 3)^2 = 25
    • D(x - 2)² + (y - 3)² = 25(x - 2)^2 + (y - 3)^2 = 25
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    Answer: (x - 2)² + (y + 3)² = 25(x - 2)^2 + (y + 3)^2 = 25

    The equation is (x - h)² + (y - k)² = r². With center (2, -3), y - (-3) becomes y + 3, and the right side is 5² = 25.The equation is (x - h)^2 + (y - k)^2 = r^2. With center (2, -3), y - (-3) becomes y + 3, and the right side is 5^2 = 25.

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34. 34Right trianglesChallenge

    A ladder 13 meters long leans against a wall with its foot 5 meters from the base. Enter how far up the wall it reaches, in meters.A ladder 13 meters long leans against a wall with its foot 5 meters from the base. Enter how far up the wall it reaches, in meters.

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

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    Answer: 1212

    The ladder is the hypotenuse, so 5² + h² = 13² gives h² = 169 - 25 = 144 and h = 12 meters.The ladder is the hypotenuse, so 5^2 + h^2 = 13^2 gives h^2 = 169 - 25 = 144 and h = 12 meters.

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35. 35PolygonsStandard

    Each interior angle of a regular polygon measures 140°. How many sides does it have?Each interior angle of a regular polygon measures 140°. How many sides does it have?

    • A99
    • B77
    • C88
    • D1010
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    Answer: 99

    Each exterior angle is 180 - 140 = 40°, and exterior angles total 360, so there are 36040 = 9 sides. Working through the exterior angle is faster than the interior-sum formula.Each exterior angle is 180 - 140 = 40°, and exterior angles total 360, so there are 360/40 = 9 sides. Working through the exterior angle is faster than the interior-sum formula.

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36. 36AreaStandard

    A rectangle has perimeter 34 and length 10. What is its area?A rectangle has perimeter 34 and length 10. What is its area?

    • A140140
    • B170170
    • C240240
    • D7070
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    Answer: 7070

    Perimeter 34 with length 10 gives 2(10) + 2w = 34, so w = 7. The area is 10 × 7 = 70.Perimeter 34 with length 10 gives 2(10) + 2w = 34, so w = 7. The area is 10 × 7 = 70.

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37. 37Trigonometric ratiosChallenge

    In a right triangle, tan A = 34. What is sin A?In a right triangle, tan A = 3/4. What is sin A?

    • A353/5
    • B454/5
    • C535/3
    • D343/4
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    Answer: 353/5

    Tangent 34 means the opposite and adjacent sides are in the ratio 3 to 4, which makes the hypotenuse 5 by the Pythagorean theorem. Sine is opposite over hypotenuse, so 35.Tangent 3/4 means the opposite and adjacent sides are in the ratio 3 to 4, which makes the hypotenuse 5 by the Pythagorean theorem. Sine is opposite over hypotenuse, so 3/5.

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38. 38VolumeFoundation

    A rectangular box measures 5 by 4 by 3 centimeters. Enter its volume in cubic centimeters.A rectangular box measures 5 by 4 by 3 centimeters. Enter its volume in cubic centimeters.

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

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    Answer: 6060

    Volume is length times width times height: 5 × 4 × 3 = 60 cubic centimeters.Volume is length times width times height: 5 × 4 × 3 = 60 cubic centimeters.

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39. 39Coordinate geometryStandard

    A line segment has one endpoint at (1, 2) and midpoint at (4, 6). What is the other endpoint?A line segment has one endpoint at (1, 2) and midpoint at (4, 6). What is the other endpoint?

    • A(5, 8)(5, 8)
    • B(3, 4)(3, 4)
    • C(2.5, 4)(2.5, 4)
    • D(7, 10)(7, 10)
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    Answer: (7, 10)(7, 10)

    The midpoint is the average of the endpoints, so the other endpoint is twice the midpoint minus the known one: (8 - 1, 12 - 2) = (7, 10).The midpoint is the average of the endpoints, so the other endpoint is twice the midpoint minus the known one: (8 - 1, 12 - 2) = (7, 10).

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40. 40AnglesFoundation

    Two angles are supplementary and one measures 47°. What is the other?Two angles are supplementary and one measures 47°. What is the other?

    • A137°137°
    • B43°43°
    • C133°133°
    • D313°313°
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    Answer: 133°133°

    Supplementary angles total 180°, so the other is 180 - 47 = 133°. Complementary angles, which total 90, would give 43.Supplementary angles total 180°, so the other is 180 - 47 = 133°. Complementary angles, which total 90, would give 43.

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41. 41Triangle anglesStandard

    In a triangle, the second angle is twice the first and the third is three times the first. What is the largest angle?In a triangle, the second angle is twice the first and the third is three times the first. What is the largest angle?

    • A60°60°
    • B30°30°
    • C120°120°
    • D90°90°
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    Answer: 90°90°

    The angles are x, 2x and 3x, totalling 6x = 180, so x = 30. The largest is 3x = 90°.The angles are x, 2x and 3x, totalling 6x = 180, so x = 30. The largest is 3x = 90°.

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42. 42CirclesChallenge

    A circle has equation (x - 3)² + (y + 1)² = 49. What is its radius?A circle has equation (x - 3)^2 + (y + 1)^2 = 49. What is its radius?

    • A24.524.5
    • B1414
    • C4949
    • D77
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    Answer: 77

    The standard form sets the right-hand side to r², so r² = 49 and r = 7. Reading 49 as the radius skips the square root.The standard form sets the right-hand side to r^2, so r^2 = 49 and r = 7. Reading 49 as the radius skips the square root.

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43. 43AreaStandard

    A circle of radius 3 is cut from a square of side 6. What area remains, in terms of π?A circle of radius 3 is cut from a square of side 6. What area remains, in terms of π?

    • A36 - 6π36 - 6π
    • B36 - 3π36 - 3π
    • C36 - 9π36 - 9π
    • D18 - 9π18 - 9π
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    Answer: 36 - 9π36 - 9π

    The square has area 36 and the circle π(3)² = 9π. The circle fits exactly inside, so the remaining area is 36 - 9π.The square has area 36 and the circle π(3)^2 = 9π. The circle fits exactly inside, so the remaining area is 36 - 9π.

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44. 44VolumeChallenge

    A cylinder and a cone have the same radius and the same height. Enter the number of cone-fulls needed to fill the cylinder.A cylinder and a cone have the same radius and the same height. Enter the number of cone-fulls needed to fill the cylinder.

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

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    Answer: 33

    A cone's volume is one third of the cylinder with the same base and height, so three cone-fulls fill it.A cone's volume is one third of the cylinder with the same base and height, so three cone-fulls fill it.

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45. 45Coordinate geometryStandard

    What is the slope of a line perpendicular to the segment joining (1, 3) and (5, 11)?What is the slope of a line perpendicular to the segment joining (1, 3) and (5, 11)?

    • A-2-2
    • B-12-1/2
    • C22
    • D121/2
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    Answer: -12-1/2

    The segment's slope is 11 - 35 - 1 = 2. A perpendicular slope is the negative reciprocal, which is -12.The segment's slope is (11 - 3)/(5 - 1) = 2. A perpendicular slope is the negative reciprocal, which is -1/2.

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46. 46Trigonometric ratiosChallenge

    In a right triangle, cos B = 513. What is tan B?In a right triangle, cos B = 5/13. What is tan B?

    • A13513/5
    • B12512/5
    • C5125/12
    • D121312/13
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    Answer: 12512/5

    Cosine 513 means the adjacent side is 5 and the hypotenuse 13, so the opposite side is 12 by the Pythagorean theorem. Tangent is opposite over adjacent, giving 125.Cosine 5/13 means the adjacent side is 5 and the hypotenuse 13, so the opposite side is 12 by the Pythagorean theorem. Tangent is opposite over adjacent, giving 12/5.

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47. 47PolygonsFoundation

    What is the measure of each exterior angle of a regular hexagon?What is the measure of each exterior angle of a regular hexagon?

    • A720°720°
    • B60°60°
    • C45°45°
    • D120°120°
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    Answer: 60°60°

    Exterior angles of any polygon total 360°. For a regular hexagon that is 3606 = 60° each. The 120-degree figure is the interior angle.Exterior angles of any polygon total 360°. For a regular hexagon that is 360/6 = 60° each. The 120-degree figure is the interior angle.

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48. 48Similar trianglesStandard

    Two similar rectangles have widths 4 and 10. If the smaller has area 24, what is the area of the larger?Two similar rectangles have widths 4 and 10. If the smaller has area 24, what is the area of the larger?

    • A240240
    • B9696
    • C150150
    • D6060
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    Answer: 150150

    The scale factor is 104 = 2.5, and area scales with its square: 24 × 6.25 = 150. Multiplying the area by 2.5 gives 60, which is the trap.The scale factor is 10/4 = 2.5, and area scales with its square: 24 × 6.25 = 150. Multiplying the area by 2.5 gives 60, which is the trap.

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49. 49AreaStandard

    A triangle has vertices at (0, 0), (6, 0) and (0, 8). What is its area?A triangle has vertices at (0, 0), (6, 0) and (0, 8). What is its area?

    • A1010
    • B4848
    • C2424
    • D1414
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    Answer: 2424

    The legs along the axes are the base and height, 6 and 8. Area is (12)(6)(8) = 24. The value 48 is the rectangle, not the triangle.The legs along the axes are the base and height, 6 and 8. Area is (1/2)(6)(8) = 24. The value 48 is the rectangle, not the triangle.

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50. 50CirclesStandard

    A sector of a circle of radius 6 has a central angle of 60°. What is its area in terms of π?A sector of a circle of radius 6 has a central angle of 60°. What is its area in terms of π?

    • A3π3π
    • B12π12π
    • C6π6π
    • D36π36π
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    Answer: 6π6π

    A 60-degree sector is one sixth of the circle. The full area is 36π, so the sector is 36π / 6 = 6π.A 60-degree sector is one sixth of the circle. The full area is 36π, so the sector is 36π / 6 = 6π.

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51. 51Right trianglesChallenge

    A rectangle measures 9 by 12. Enter the length of its diagonal.A rectangle measures 9 by 12. Enter the length of its diagonal.

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

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    Answer: 1515

    The diagonal is the hypotenuse of a right triangle with legs 9 and 12: √81 + 144 = √225 = 15.The diagonal is the hypotenuse of a right triangle with legs 9 and 12: √(81 + 144) = √(225) = 15.

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52. 52VolumeStandard

    A cylinder has volume 100π and height 4. What is its radius?A cylinder has volume 100π and height 4. What is its radius?

    • A2525
    • B55
    • C1010
    • D6.256.25
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    Answer: 55

    From π r² h = 100π with h = 4: r² = 25, so r = 5. Stopping at 25 skips the square root.From π r^2 h = 100π with h = 4: r^2 = 25, so r = 5. Stopping at 25 skips the square root.

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53. 53AnglesStandard

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

    • A30°30°
    • B45°45°
    • C120°120°
    • D60°60°
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    Answer: 60°60°

    Complementary angles total 90°, so x + 2x = 90 gives x = 30 and the larger is 60. Using 180 would make them supplementary instead.Complementary angles total 90°, so x + 2x = 90 gives x = 30 and the larger is 60. Using 180 would make them supplementary instead.

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54. 54Similar trianglesChallenge

    A 2-meter pole casts a 3-meter shadow. At the same moment a building casts a 42-meter shadow. Enter the building's height in meters.A 2-meter pole casts a 3-meter shadow. At the same moment a building casts a 42-meter shadow. Enter the building's height in meters.

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

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    Answer: 2828

    Height over shadow is constant in similar right triangles: 23 of 42 is 28 meters.Height over shadow is constant in similar right triangles: 2/3 of 42 is 28 meters.

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55. 55Trigonometric ratiosStandard

    In a right triangle with the right angle at C, side a = 6 and side b = 8. What is sin A, where a is opposite angle A?In a right triangle with the right angle at C, side a = 6 and side b = 8. What is sin A, where a is opposite angle A?

    • A0.750.75
    • B0.80.8
    • C1.251.25
    • D0.60.6
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    Answer: 0.60.6

    The hypotenuse is √36 + 64 = 10. Sine is the opposite side over the hypotenuse: 610 = 0.6.The hypotenuse is √(36 + 64) = 10. Sine is the opposite side over the hypotenuse: 6/10 = 0.6.

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56. 56Coordinate geometryStandard

    Which point lies on the line y = 2x - 5?Which point lies on the line y = 2x - 5?

    • A(2, 1)(2, 1)
    • B(4, 3)(4, 3)
    • C(3, 4)(3, 4)
    • D(0, 5)(0, 5)
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    Answer: (4, 3)(4, 3)

    Substituting x = 4 gives y = 8 - 5 = 3, so (4, 3) is on the line. Checking (2, 1) gives y = -1, not 1.Substituting x = 4 gives y = 8 - 5 = 3, so (4, 3) is on the line. Checking (2, 1) gives y = -1, not 1.

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