Trigonometry Right Triangle Solutions 3 — Questions and Answers
Question 1: In a right triangle with a 45° angle, if the hypotenuse is 10, what are the lengths of the two legs?
- 5√2 each (Correct answer)
- 5 each
- 10√2 each
- 10/√3 each
Correct answer: 5√2 each
A 45-45-90 triangle has legs = hypotenuse/√2 = 10/√2 = 5√2.
Question 2: Right triangle XYZ has the right angle at Z. If XY = 17 and YZ = 15, what is tan X?
- 15/8 (Correct answer)
- 8/15
- 15/17
- 8/17
Correct answer: 15/8
XZ = √(17² − 15²) = √(289 − 225) = √64 = 8; tan X = opposite/adjacent = YZ/XZ = 15/8.
Question 3: An observer stands 50 m from a building and looks up at a 38° angle of elevation to the top. What is the approximate height of the building?
- 50 tan 38° (Correct answer)
- 50 sin 38°
- 50 cos 38°
- 50 / tan 38°
Correct answer: 50 tan 38°
The height = adjacent × tan(angle) = 50 tan 38°.
Question 4: In a 30-60-90 triangle, the side opposite the 60° angle is 6√3. What is the hypotenuse?
- 12 (Correct answer)
- 6
- 6√2
- 12√3
Correct answer: 12
The side opposite 60° is (√3/2) × hypotenuse, so hypotenuse = 6√3 / (√3/2) = 12.
Question 5: If cos θ = 5/13 in a right triangle, what is tan θ?
- 12/5 (Correct answer)
- 5/12
- 13/5
- 5/13
Correct answer: 12/5
With adjacent = 5 and hypotenuse = 13, opposite = 12 (Pythagorean triple), so tan θ = 12/5.
Question 6: A right triangle has legs a and b and hypotenuse c. If a = b and c = 18, what is a?
- 9√2 (Correct answer)
- 9
- 18√2
- 6√3
Correct answer: 9√2
Since a = b and a² + b² = 18², we get 2a² = 324, so a = √162 = 9√2.
Question 7: In a right triangle, the hypotenuse is 26 and one acute angle is 67°. What is the length of the side opposite the 67° angle?
- 26 sin 67° (Correct answer)
- 26 cos 67°
- 26 tan 67°
- 26 / sin 67°
Correct answer: 26 sin 67°
Opposite side = hypotenuse × sin(67°) = 26 sin 67°.
Trigonometry Right Triangle Solutions 3
In a right triangle with a 45° angle, if the hypotenuse is 10, what are the lengths of the two legs?