Question

determining trigonometric ratios of similar triangles
consider δabc ~ δxyz.
what is the value of tan(a)?
what is the value of tan(x)?
what is true about the two tangent ratios?

Explanation:

Step1: Recall tan definition for ∠A

For right triangle $\Delta ABC$, $\tan(A) = \frac{\text{opposite side to } A}{\text{adjacent side to } A}$
$\tan(A) = \frac{BC}{AC} = \frac{3}{4}$

Step2: Recall tan definition for ∠X

For right triangle $\Delta XYZ$, $\tan(X) = \frac{\text{opposite side to } X}{\text{adjacent side to } X}$
$\tan(X) = \frac{YZ}{XZ} = \frac{6}{8} = \frac{3}{4}$

Step3: Compare the two ratios

Since $\tan(A)=\frac{3}{4}$ and $\tan(X)=\frac{3}{4}$, the ratios are equal.

Answer:

1. $\tan(A) = \frac{3}{4}$
2. $\tan(X) = \frac{3}{4}$
3. The two tangent ratios are equal.