Triangle PQR is similar to triangle STU, where P, Q, and R... | SAT Lines, angles, and triangles Question, Answer Explained

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Math · Geometry and TrigonometryMedium

Triangle $PQR$ is similar to triangle $STU$ , where $P$ , $Q$ , and $R$ correspond to $S$ , $T$ , and $U$ , respectively. In triangle $PQR$ , $PQ = 8$ and $QR = 14$ . In triangle $STU$ , $ST = 12$ , and the perimeter of triangle $STU$ is $51$ . What is the length of $\overline{PR}$ ?

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Answer: 12

Side $\overline{PQ}$ corresponds to side $\overline{ST}$ , so the scale factor from triangle $STU$ to triangle $PQR$ is $\frac{8}{12} = \frac{2}{3}$ . Perimeters of similar triangles have the same ratio, so the perimeter of triangle $PQR$ is $\frac{2}{3}(51) = 34$ . Then $PR = 34 - 8 - 14 = 12$ .

Subtracting $8$ and $14$ from $51$ gives $29$ , but $51$ is the perimeter of triangle $STU$ , not triangle $PQR$ .

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