In right triangle PQR, the right angle is located at vertex Q. The length of leg PQ is 30 … — ACT

In right triangle $PQR$ , the right angle is located at vertex $Q$ . The length of leg $PQ$ is $30$ centimeters. If $\sin(P) = \frac{8}{17}$ , what is the length, in centimeters, of leg $QR$ ?

$16$ Answer
B
$\frac{240}{17}$
C
$18$
D
$34$
E
$\frac{225}{4}$

Answer

16 centimeters

In right triangle

PQR

\sin(P) = \frac{QR}{PR} = \frac{8}{17}

. The corresponding cosine ratio is

\cos(P) = \frac{PQ}{PR} = \frac{15}{17}

. Given

PQ = 30

, we solve

\frac{30}{PR} = \frac{15}{17}

to find

PR = 34

. Then, leg

QR = 34 \times \frac{8}{17} = 16

Step-by-Step Solution

Express the given sine ratio in terms of the triangle sides.

\sin(P) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR} = \frac{8}{17}

By definition of the sine function in a right triangle.

Find the cosine ratio

\cos(P)

using the Pythagorean identity or the 8-15-17 right triangle ratio.

\cos(P) = \sqrt{1 - \sin^2(P)} = \sqrt{1 - \frac{64}{289}} = \frac{15}{17}

Cosine represents the ratio of the adjacent side (

PQ

) to the hypotenuse (

PR

Calculate the hypotenuse

PR

using the known leg

PQ = 30

\frac{PQ}{PR} = \frac{15}{17} \implies \frac{30}{PR} = \frac{15}{17} \implies PR = 34

Setting the adjacent side ratio equal to

\cos(P)

solves for the hypotenuse length.

Calculate the opposite leg

QR

QR = PR \times \sin(P) = 34 \times \frac{8}{17} = 16

Multiplying the hypotenuse by

\sin(P)

gives the opposite leg length.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)

Estimated Time:1m 15s

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