Question

given $\triangle abc \sim \triangle xyz$, what is the value of $\cos(z)$?
$\bigcirc \\ \frac{5}{13}$
$\bigcirc \\ \frac{5}{12}$
$\bigcirc \\ \frac{12}{13}$
$\bigcirc \\ \frac{12}{5}$

Explanation:

Step1: Identify corresponding angles

Since $\triangle ABC \sim \triangle XYZ$, corresponding angles are congruent: $\angle Z \cong \angle C$.

Step2: Recall cosine definition for right triangles

For $\angle C$ in right $\triangle ABC$, $\cos(\theta)=\frac{\text{adjacent side}}{\text{hypotenuse}}$.

Step3: Match sides for $\angle C$

Adjacent side to $\angle C$ is $12$, hypotenuse is $13$.
<Expression>
$\cos(C)=\frac{12}{13}$
</Expression>

Step4: Equate to $\cos(Z)$

Since $\angle Z \cong \angle C$, $\cos(Z)=\cos(C)$.

Answer:

$\boldsymbol{\frac{12}{13}}$ (Option C)