Question

triangle bac was dilated from triangle bde at a scale factor of 2. what proportion proves that cos∠d = cos∠a?

Explanation:

Step1: Recall cosine - ratio definition

The cosine of an angle in a right - triangle is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. In right - triangle $BDE$, for $\angle D$, $\cos D=\frac{BD}{DE}$. In right - triangle $BAC$, for $\angle A$, $\cos A=\frac{BA}{AC}$. Since $\triangle BDE$ is dilated to $\triangle BAC$ with a scale factor of 2, the corresponding sides are in proportion. If $BD = 2$, $DE=3.61$, $BA = 4$, and $AC = 7.22$ (because of the scale - factor of 2 relationship: if the scale factor is 2, then the sides of $\triangle BAC$ are twice the sides of $\triangle BDE$).

Step2: Calculate the cosine ratios

$\cos D=\frac{BD}{DE}=\frac{2}{3.61}$ and $\cos A=\frac{BA}{AC}=\frac{4}{7.22}=\frac{2\times2}{2\times3.61}=\frac{2}{3.61}$.

Answer:

$\frac{2}{3.61}=\frac{4}{7.22}$