A coordinate grid contains a right triangle, XYZ, with the right angle located at vertex Z… — ACT

A coordinate grid contains a right triangle, $XYZ$ , with the right angle located at vertex $Z$ . The horizontal leg $XZ$ has a length of $24$ units, and the vertical leg $YZ$ has a length of $7$ units. What is the value of $\cos(X)$ ?

A
$\frac{7}{25}$
$\frac{24}{25}$ Answer
C
$\frac{7}{24}$
D
$\frac{24}{7}$
E
$\frac{25}{24}$

Answer

The correct answer is the option representing

\frac{24}{25}

The cosine of an angle in a right triangle is the ratio of the length of the adjacent leg to the length of the hypotenuse. The hypotenuse of the triangle is calculated as

25

units using the Pythagorean theorem. Relative to angle

X

, the adjacent leg is

XZ = 24

and the hypotenuse is

XY = 25

. Therefore,

\cos(X) = \frac{24}{25}

Step-by-Step Solution

Use the Pythagorean theorem to calculate the hypotenuse

XY

XY = \sqrt{XZ^2 + YZ^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25

To calculate any primary trigonometric ratio, we need the lengths of the relevant sides, including the hypotenuse.

Identify the adjacent side relative to angle

X

The leg adjacent to angle

X

XZ = 24

Angle

X

is formed by the hypotenuse

XY

and the adjacent leg

XZ

Apply the definition of cosine (adjacent over hypotenuse).

\cos(X) = \frac{XZ}{XY} = \frac{24}{25}

By SOHCAHTOA, cosine is the ratio of the adjacent side length to the hypotenuse length.

Key Concept

Using SOHCAHTOA definitions to find trigonometric ratios in a right triangle.

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