Finding Angles with the Pythagorean Theorem + Ratios

  • 10th Grade
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Cierra Henderson, MBA |
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 11 | Questions: 20 | Updated: Dec 10, 2025
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Question 1 / 21
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1) In a right triangle, the legs are 5 and 12. Find the angle θ opposite the shorter leg.

Explanation

θ = arctan(5/12) = 22.62° → 22.6°.

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About This Quiz
Finding Angles With The Pythagorean Theorem + Ratios - Quiz

Want to master both right triangle formulas and trigonometric reasoning? In this quiz, you’ll combine the Pythagorean Theorem and trig ratios to solve for missing sides and angles. You’ll tackle real-world examples — from leaning ladders to rising ramps — and learn how geometry and trigonometry work together. By the... see moreend, you’ll be confident solving any right triangle with precision and understanding!
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2) A right triangle has legs 12 and 35. Find the hypotenuse.

Explanation

Hypotenuse = √(12²+35²) = √1369 = 37.

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3) A ladder 15 ft long leans against a wall. The base is 9 ft from the wall. Find the approximate angle of elevation.

Explanation

cos θ = 9/15 = 0.6 → θ = arccos(0.6) = 53.13° ≈ 53°.

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4) A right triangle has opposite = 5, adjacent = 12. Find θ to the nearest degree.

Explanation

θ = arctan(5/12) = 22.62° → 23°.

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5) A ramp rises 4 m over a run of 12 m. Find the angle of elevation to the nearest degree.

Explanation

θ = arctan(4/12) = arctan(1/3) = 18.43° → 18°.

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6) A right triangle has hypotenuse 25 and one leg 20. Find the other leg.

Explanation

Other leg = √(25² − 20²) = √(625 − 400) = √225 = 15.

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7) A right triangle has legs 9 and 12. Find the hypotenuse.

Explanation

Hypotenuse = √(9²+12²) = √225 = 15.

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8) A right triangle has opposite = 9, hypotenuse = 41. Find θ to the nearest degree.

Explanation

θ = arcsin(9/41) = 12.68° → 13°.

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9) A ladder 20 ft long makes a 60° angle with the ground. How high does it reach?

Explanation

Height = 20·sin60° = 20·(√3/2) = 10√3 ft ≈ 17.32 ft.

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10) A right triangle has adjacent = 14, opposite = 9. Find the hypotenuse to the nearest tenth.

Explanation

Hypotenuse = √(14²+9²) = √277 = 16.64 ≈ 16.6.

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11) A right triangle has hypotenuse 34 and one leg 30. Find the other leg.

Explanation

Other leg = √(34² − 30²) = √(1156 − 900) = √256 = 16.

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12) A right triangle has opposite = 18, adjacent = 24. Find θ to the nearest degree.

Explanation

θ = arctan(18/24) = 36.87° → 37°.

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13) A right triangle has opposite side = 7 and hypotenuse = 25. Find θ to the nearest degree.

Explanation

θ = arcsin(7/25) = 16.26° → 16°.

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14) A right triangle has legs 11 and 60. Find the hypotenuse.

Explanation

Hypotenuse = √(11²+60²) = √3721 = 61.

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15) A right triangle has opposite = 20, adjacent = 21. Find θ to the nearest degree.

Explanation

θ = arctan(20/21) = 43.60° → 44°.

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16) A ladder 10 m long leans so it reaches 8 m high. Find the angle with the ground rounded to the nearest degree.

Explanation

sin θ = 8/10 → θ = 53.13° → 53°.

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17) A right triangle has adjacent = 9, hypotenuse = 41. Find θ to the nearest degree.

Explanation

θ = arccos(9/41) = 77.32° → 77°.

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18) A right triangle has opposite side = 24 and adjacent side = 7. Find θ to the nearest degree.

Explanation

θ = arctan(24/7) = 73.74° → 74°.

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19) A right triangle has opposite = 12, adjacent = 5. Find θ to the nearest degree.

Explanation

θ = arctan(12/5) = 67.38° → ≈ 67.4°, nearest among options is 68°.

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20) A ladder 18 ft long leans so it makes a 65° angle with the ground. Approximately, how high does it reach?

Explanation

Height = 18·sin65° = 16.31 ft → 16.3 ft.

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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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In a right triangle, the legs are 5 and 12. Find the angle θ opposite...
A right triangle has legs 12 and 35. Find the hypotenuse.
A ladder 15 ft long leans against a wall. The base is 9 ft from the...
A right triangle has opposite = 5, adjacent = 12. Find θ to the...
A ramp rises 4 m over a run of 12 m. Find the angle of elevation to...
A right triangle has hypotenuse 25 and one leg 20. Find the other leg.
A right triangle has legs 9 and 12. Find the hypotenuse.
A right triangle has opposite = 9, hypotenuse = 41. Find θ to the...
A ladder 20 ft long makes a 60° angle with the ground. How high does...
A right triangle has adjacent = 14, opposite = 9. Find the hypotenuse...
A right triangle has hypotenuse 34 and one leg 30. Find the other leg.
A right triangle has opposite = 18, adjacent = 24. Find θ to the...
A right triangle has opposite side = 7 and hypotenuse = 25. Find θ to...
A right triangle has legs 11 and 60. Find the hypotenuse.
A right triangle has opposite = 20, adjacent = 21. Find θ to the...
A ladder 10 m long leans so it reaches 8 m high. Find the angle with...
A right triangle has adjacent = 9, hypotenuse = 41. Find θ to the...
A right triangle has opposite side = 24 and adjacent side = 7. Find θ...
A right triangle has opposite = 12, adjacent = 5. Find θ to the...
A ladder 18 ft long leans so it makes a 65° angle with the ground....
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