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Problem 131

Geometry Difficulty 1.4 Multiple choice CEMC Gauss (Grade 8) · Canada · 2017

Line segments $PQ$ and $RS$ intersect at $T$ , as shown.

If $TS=TQ$ and $\angle PTR=88^{\circ}$ , the value of $x$ is

Pick one

Official solution

Each of the 4 smaller triangles is equilateral and thus has sides of equal length.

Each of these smaller triangles has a perimeter of 9 cm and so has sides of length $\frac93=3$ cm.

In $\triangle PQR$ , side $PQ$ is made up of two such sides of length 3 cm and thus $PQ=2\times3=6$ cm.

Since $\triangle PQR$ is equilateral, then $PR=QR=PQ=6$ cm.

Therefore, the perimeter of $\triangle PQR$ is $3\times6=18$ cm.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.