CEMC Cayley

Geometry Difficulty 2.8 Junior Find the answer

What is the length of $SR$ if in $\triangle PQR$ , $PS$ is perpendicular to $QR$ , $RT$ is perpendicular to $PQ$ , $PT=1$ , $TQ=4$ , and $QS=3$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $PT=1$ and $TQ=4$ , then $PQ=PT+TQ=1+4=5$ . $\triangle PSQ$ is right-angled at $S$ and has hypotenuse $PQ$ . By the Pythagorean Theorem, $PS^{2}=PQ^{2}-QS^{2}=5^{2}-3^{2}=16$ . Since $PS>0$ , then $PS=4$ . Consider $\triangle PSQ$ and $\triangle RTQ$ . These triangles are similar, so $\frac{PQ}{QS}=\frac{QR}{TQ}$ . Thus, $\frac{5}{3}=\frac{QR}{4}$ and $QR=\frac{20}{3}$ . Finally, $SR=QR-QS=\frac{20}{3}-3=\frac{11}{3}$ .

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