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The diagram shows that triangle $XYZ$ may be divided into 9 congruent triangles. The square $PQRS$ is made up of 4 of these 9 triangles. Therefore, the ratio is $4:9$.


                                 Subdivided right-angled triangle circumscribing square

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.