CEMC Cayley, 2015

Geometry Difficulty 2.5 Junior Find the answer Canada

In the diagram, $\triangle PQR$ is isosceles with $PQ=PR$ and $\triangle PRS$ is isosceles with $PS=SR=x$ .

Also, the perimeter of $\triangle PQR$ is 22, the perimeter of $\triangle PRS$ is 22, and the perimeter of $PQRS$ is 24. What is the value of $x$ ?

Pick one

Solution

Since $PS=SR=x$ and the perimeter of $\triangle PRS$ is 22, then $PR=22-PS-SR=22-2x$ .

Since $PQ=PR$ and $PR=22-2x$ , then $PQ=22-2x$ .

Since $\triangle PQR$ has perimeter 22, then $RQ=22-PR-PQ = 22-(22-2x)-(22-2x)=4x-22$ .

Since the perimeter of $PQRS$ is 24, then $\begin{aligned} PQ+RQ+SR+PS & = 24 \\ (22-2x)+(4x-22)+x+x & = 24 \\ 4x & = 24 \\ x & = 6\end{aligned}$ Therefore, $x=6$ .

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