In a right triangle, the hypotenuse has length 26, and the... | SAT Right triangles and trigonometry Question, Answer Explained

Question bank / Right triangles and trigonometry

Math · Geometry and TrigonometryMedium

In a right triangle, the hypotenuse has length $26$ , and the sine of one of the acute angles is $\frac{5}{13}$ . What is the area of the triangle?

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Answer: 120

If $\sin \theta = \frac{5}{13}$ , the leg opposite $\theta$ has length $\frac{5}{13}(26) = 10$ . By the Pythagorean theorem, the other leg has length $\sqrt{26^2 - 10^2} = \sqrt{576} = 24$ . The area is $\frac{1}{2}(10)(24) = 120$ .

Using leg lengths $5$ and $12$ gives an area of $30$ , but this triangle's hypotenuse is $26$ , so its sides are twice as long and its area is $4$ times as large.

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