Question

what is the length of \\( \overline { f r } \\)?

Explanation:

Step1: Identify similar triangles

Since \(\angle D=\angle P = 25^{\circ}\), \(\angle DEF=\angle PQR=114^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle DEF\sim\triangle PQR\).

Step2: Set up the proportion

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 let \(PR=x\). Substituting the values into the proportion: \(\frac{15}{5}=\frac{21}{x}\).

Step3: Solve the proportion for \(x\)

Cross - multiply: \(15x=5\times21\).
Simplify the right - hand side: \(15x = 105\).
Divide both sides by 15: \(x=\frac{105}{15}=7\).

Answer:

\(7\)