Question

given $\triangle xyzsim\triangle pqr$, if $angle x = 70^{circ},angle p = 70^{circ}$, and the ratios $\frac{xy}{pq}=\frac{xz}{pr}=2$, which of the following is true?
a. only $\triangle pqr$ is larger than $\triangle xyz$
b. the angles are not proportional
c. the triangles are similar with a scale factor of 2
d. the triangles are congruent

Explanation:

Step1: Recall similarity definition

Two triangles $\triangle XYZ\sim\triangle PQR$ means their corresponding angles are equal and corresponding - side lengths are in proportion. Given $\frac{XY}{PQ}=\frac{XZ}{PR} = 2$, this is the scale - factor of the similarity.

Step2: Analyze each option

• Option a: Since $\frac{XY}{PQ}=\frac{XZ}{PR}=2$, $\triangle XYZ$ is larger than $\triangle PQR$, not the other way around.
• Option b: For similar triangles, corresponding angles are equal, so they are proportional (in a sense of being equal).
• Option c: Given $\triangle XYZ\sim\triangle PQR$ and $\frac{XY}{PQ}=\frac{XZ}{PR}=2$, the triangles are similar with a scale factor of 2. This is correct.
• Option d: For congruent triangles, the scale factor should be 1. Here the scale factor is 2, so they are not congruent.

Answer:

C. The triangles are similar with a scale factor of 2