find the cosine of $angle g$.\nsimplify your answer and write it as a proper fraction, improper fraction, or…

find the cosine of $angle g$.\nsimplify your answer and write it as a proper fraction, improper fraction, or whole number.\n$cos(g) = square$

find the cosine of $angle g$.\nsimplify your answer and write it as a proper fraction, improper fraction, or whole number.\n$cos(g) = square$

Answer

Explanation:

Step1: Identify similar triangles

Triangles $BCD$ and $GEF$ are right triangles with congruent acute angles ($\angle B \cong \angle G$), so they are similar. Corresponding sides are proportional, so the cosine of $\angle G$ equals the cosine of $\angle B$.

Step2: Recall cosine definition

For acute $\angle B$ in right $\triangle BCD$, $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step3: Substitute side lengths

Adjacent side to $\angle B$ is $BC=40$, hypotenuse is $BD=58$. $\cos(G)=\cos(B)=\frac{40}{58}$

Step4: Simplify the fraction

Divide numerator and denominator by 2: $\frac{40\div2}{58\div2}=\frac{20}{29}$

Answer:

$\frac{20}{29}$