Question

given △xyz∼△pqr, if ∠x=70°, ∠p=70°, and the sides
xy/xz = pq/pr = 2, which of the following is true?
o a. the angles are not proportional
o b. the triangles are similar with a scale factor of 2
o c. only △pqr is larger than △xyz
o d. the triangles are congruent

Explanation:

Step1: Recall properties of similar triangles

If \(\triangle XYZ\sim\triangle PQR\), then corresponding angles are equal (\(\angle X=\angle P\), \(\angle Y = \angle Q\), \(\angle Z=\angle R\)) and the ratios of corresponding sides are equal (i.e., \(\frac{XY}{PQ}=\frac{YZ}{QR}=\frac{XZ}{PR}\)).

Step2: Analyze each option

• Option a:

Since \(\triangle XYZ\sim\triangle PQR\), by the property of similar triangles, the ratios of corresponding sides are equal. So, \(\frac{XY}{PQ}=\frac{YZ}{QR}=\frac{XZ}{PR}\), which means the sides are proportional. So, the statement “The angles are not proportional” is false.

• Option b:

Given \(\frac{XY}{PQ}=\frac{YZ}{QR}=\frac{XZ}{PR} = 2\). When the ratio of corresponding sides of two similar triangles \(k=\frac{side\ of\ \triangle XYZ}{side\ of\ \triangle PQR}=2\), it means \(\triangle XYZ\) is larger than \(\triangle PQR\) (not the other way around). The scale - factor formula is \(k=\frac{\text{side of first triangle}}{\text{side of second triangle}}\). If \(k > 1\), the first triangle is larger. Here, if we consider \(\frac{XY}{PQ}=2\), \(\triangle XYZ\) is larger. So, the statement “The triangles are similar with a scale factor of 2” is correct.

• Option c:

Since \(\frac{XY}{PQ}=\frac{YZ}{QR}=\frac{XZ}{PR}=2\), \(\triangle XYZ\) is larger than \(\triangle PQR\) (because \(XY = 2PQ\), \(YZ=2QR\), \(XZ = 2PR\)). So, the statement “Only \(\triangle PQR\) is larger than \(\triangle XYZ\)” is false.

• Option d:

For congruent triangles, the ratio of corresponding sides \(k = 1\). Here, \(k = 2\) (since \(\frac{XY}{PQ}=2\)), so the triangles are not congruent. So, the statement “The triangles are congruent” is false.

Answer:

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