In right triangle XYZ, the right angle is located at vertex Y. The hypotenuse XZ has a len… — ACT

In right triangle $XYZ$ , the right angle is located at vertex $Y$ . The hypotenuse $XZ$ has a length of $39$ inches. If $\cos(X) = \frac{5}{13}$ , what is the length, in inches, of leg $YZ$ ?

A
15
B
24
36Answer
D
42
E
65

Answer

36 inches

By SOH CAH TOA,

\cos(X) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{XY}{39}

. Setting

\frac{XY}{39} = \frac{5}{13}

gives

XY = 15

inches. Applying the Pythagorean theorem to find the remaining leg

YZ

gives

YZ = \sqrt{39^2 - 15^2} = \sqrt{1296} = 36

inches. Alternatively, recognizing the ratio

\cos(X) = \frac{5}{13}

implies

\sin(X) = \frac{12}{13}

for a 5-12-13 right triangle, so

YZ = 39 \times \frac{12}{13} = 36

inches.

Step-by-Step Solution

Use the definition of cosine (SOH CAH TOA) to find the length of leg

XY

, which is adjacent to angle

X

\cos(X) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{XY}{XZ} \implies \frac{5}{13} = \frac{XY}{39} \implies XY = 39 \times \frac{5}{13} = 15

inches.

Cosine is defined as the ratio of the adjacent side to the hypotenuse.

Apply the Pythagorean theorem (

XY^2 + YZ^2 = XZ^2

) to calculate the length of leg

YZ

15^2 + YZ^2 = 39^2 \implies 225 + YZ^2 = 1521 \implies YZ^2 = 1296 \implies YZ = \sqrt{1296} = 36

inches.

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) and Pythagorean Theorem

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