Question

question 9 of 10
what is the length of \\( \overline { p r } \\) ?
a. 7
b. 11
c. 9
d. 63

Explanation:

Step1: Determine similarity of triangles

Since two angles of \(\triangle DEF\) (\(25^{\circ}\) and \(114^{\circ}\)) are equal to two angles of \(\triangle PQR\) (\(25^{\circ}\) and \(114^{\circ}\)), by the AA (Angle - Angle) similarity criterion, \(\triangle DEF\sim\triangle PQR\).

Step2: Set up proportion for similar triangles

For similar triangles \(\triangle DEF\) and \(\triangle PQR\), the ratios of corresponding sides are equal. That is \(\frac{DE}{PQ}=\frac{DF}{PR}\). We know \(DE = 15\), \(PQ=5\), \(DF = 21\), and \(PR=x\). Substituting the values into the proportion: \(\frac{15}{5}=\frac{21}{x}\).

Step3: Solve the proportion

Cross - multiply the proportion \(15x=21\times5\). Then \(15x = 105\). Divide both sides by 15: \(x=\frac{105}{15}=7\).

Answer:

A. 7