Question

if $cos f^circ = \frac{4}{9}$ and the measure of $overline{xw}$ is 16 units, what is the measure of $overline{xy}$?

Explanation:

Step1: Identify the right triangle and cosine definition

In right triangle \( XWY \) (right - angled at \( W \)), for angle \( f^{\circ} \) (at vertex \( X \)), the cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. The adjacent side to angle \( f^{\circ} \) is \( XW \), and the hypotenuse is \( XY \). So, \( \cos f^{\circ}=\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{XW}{XY} \).

Step2: Substitute the known values

We know that \( \cos f^{\circ}=\frac{4}{9} \) and \( XW = 16 \) units. Substituting these values into the cosine formula: \( \frac{4}{9}=\frac{16}{XY} \).

Step3: Solve for \( XY \)

Cross - multiply to solve for \( XY \). Cross - multiplying gives us \( 4\times XY=16\times9 \). Then, \( 4XY = 144 \). Divide both sides by 4: \( XY=\frac{144}{4}=36 \).

Answer:

The measure of \( \overline{XY} \) is 36 units.