In right triangle XYZ, the right angle is at vertex Y, and line segment YW is an altitude … — ACT

In right triangle $XYZ$ , the right angle is at vertex $Y$ , and line segment $YW$ is an altitude perpendicular to hypotenuse $XZ$ at point $W$ . If the length of side $XY$ is $15$ units and $\cos(X) = \frac{4}{5}$ , what is the length of line segment $ZW$ ?

A
$9$
B
$12$
$\frac{27}{4}$ Answer
D
$\frac{45}{4}$
E
$16$

Answer

The length of line segment

ZW

\frac{27}{4}

units.

In right triangle

XYZ

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

. Given

XY = 15

and

\cos(X) = \frac{4}{5}

, we solve for hypotenuse

XZ = \frac{75}{4}

. In right triangle

XYW

\cos(X) = \frac{XW}{XY} = \frac{XW}{15}

, yielding

XW = 12

. Subtracting

XW

from total hypotenuse

XZ

yields

ZW = \frac{75}{4} - 12 = \frac{27}{4}

Step-by-Step Solution

Find the length of hypotenuse

XZ

using

\cos(X)

\triangle XYZ

XZ = \frac{75}{4}

\triangle XYZ

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

. Substituting

\cos(X) = \frac{4}{5}

and

XY = 15

gives

\frac{4}{5} = \frac{15}{XZ}

, so

XZ = \frac{15 \times 5}{4} = \frac{75}{4}

Find the length of segment

XW

using

\cos(X)

in right triangle

XYW

XW = 12

In right triangle

XYW

(with right angle at

W

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

. Substituting

\cos(X) = \frac{4}{5}

and

XY = 15

gives

\frac{4}{5} = \frac{XW}{15}

, so

XW = 12

Calculate segment

ZW

by subtracting

XW

from total hypotenuse

XZ

ZW = \frac{27}{4}

Since point

W

lies on segment

XZ

ZW = XZ - XW = \frac{75}{4} - 12 = \frac{75}{4} - \frac{48}{4} = \frac{27}{4}

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) in Nested Right Triangles

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