Question

consider △abc ~ △xyz. what is the value of tan(a)? 3/4 what is the value of tan(x)? dropdown is true about the two tangent ratios? dropdown with options 3/4, 3/5, 4/5 (image shows two right triangles: △abc with right angle at c, ac=4, bc=3, ab=5; △xyz with right angle at z, xz=8, yz=6, xy=10)

Explanation:

For $\tan(X)$:

Step1: Recall tangent definition

In a right triangle, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Identify sides for $\angle X$

In $\triangle XYZ$, $\angle Z$ is right. Opposite to $\angle X$ is $YZ = 6$, adjacent is $XZ = 8$.

Step3: Calculate $\tan(X)$

$\tan(X)=\frac{YZ}{XZ}=\frac{6}{8}=\frac{3}{4}$.

For the relationship of the two tangent ratios:

Since $\triangle ABC \sim \triangle XYZ$ (similar triangles), corresponding angles are equal. So $\angle A=\angle X$, and thus their tangent ratios are equal.

Answer:

s:

• $\tan(X)$: $\frac{3}{4}$
• Relationship: The two tangent ratios are equal (both $\frac{3}{4}$)