In right triangle XYZ, the measure of angle Y is 90^\circ. If the length of side XY is 12 … — SAT

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ . If the length of side $XY$ is $12$ and the area of the triangle is $30$ , what is the value of $\cos(X)$ ?

$\frac{12}{13}$ Answer
B
$\frac{5}{13}$
C
$\frac{5}{12}$
D
$\frac{24}{25}$

Answer

\frac{12}{13}

The correct answer is the fraction expressing twelve thirteenths. The area of thirty and the leg length of twelve imply that the other leg has a length of five. Applying the Pythagorean theorem to legs of twelve and five yields a hypotenuse of thirteen. Because cosine is defined as the ratio of the adjacent leg to the hypotenuse, the cosine of angle

X

is twelve over thirteen.

Step-by-Step Solution

Find the length of leg

YZ

using the area formula.

YZ = 5

The area of a right triangle is

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Here,

\text{Area} = 30

and base

XY = 12

, so

30 = \frac{1}{2} \times 12 \times YZ \implies 30 = 6 \times YZ \implies YZ = 5

Calculate the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = 13

In right triangle

XYZ

, the Pythagorean theorem states that

XZ^2 = XY^2 + YZ^2

. Substituting the known leg lengths,

XZ^2 = 12^2 + 5^2 = 144 + 25 = 169

. Taking the square root gives

XZ = 13

Determine the value of

\cos(X)

using the trigonometric definition.

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

By definition, the cosine of an acute angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For angle

X

, the adjacent side is

XY

and the hypotenuse is

XZ

, so

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

Key Concept

Using the area formula, Pythagorean theorem, and trigonometric ratios to solve right triangles.

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