Geometry and TrigonometryVery hardLeg length from a cosine and the hypotenuse
Geometry and Trigonometry practice question
In a right triangle, the hypotenuse has length 26 and one acute angle θ satisfies cos θ = 5/13. What is the length of the leg opposite θ?
Answer: 24
cos θ = adjacent/26 = 5/13 gives adjacent = 10. Then the opposite leg is √(26² − 10²) = √(676 − 100) = √576 = 24. (A 5-12-13 triangle scaled by 2.)
Common wrong answers
- If you answered 10
- 10 is the leg ADJACENT to θ, not the opposite one.
- If you answered 12
- 12 is sin θ's numerator (12/13) without scaling to the hypotenuse of 26.
- If you answered 16
- 16 is 26 − 10, subtracting lengths instead of squares.
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