Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 2.7 Junior Find the answer

If $\triangle PQR$ is right-angled at $P$ with $PR=12$ , $SQ=11$ , and $SR=13$ , what is the perimeter of $\triangle QRS$ ?

Your answer

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

Solution

By the Pythagorean Theorem in $\triangle PRS$ , $PS=\sqrt{RS^{2}-PR^{2}}=\sqrt{13^{2}-12^{2}}=\sqrt{169-144}=\sqrt{25}=5$ since $PS>0$ . Thus, $PQ=PS+SQ=5+11=16$ . By the Pythagorean Theorem in $\triangle PRQ$ , $RQ=\sqrt{PR^{2}+PQ^{2}}=\sqrt{12^{2}+16^{2}}=\sqrt{144+256}=\sqrt{400}=20$ since $RQ>0$ . Therefore, the perimeter of $\triangle QRS$ is $RS+SQ+RQ=13+11+20=44$ .

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