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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Isosceles Reduction

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Triangles $PRS$ and $QPR$ are similar because $\angle PSR = \angle QRP$ (since $PR =PS$) and $\angle PRS = \angle QPR$ (since $QP =QR$).

Hence $\frac{SR}{RP} = \frac{RP}{PQ}$, that is $\frac{SR}{6} = \frac{6}{9}$, that is $SR = 4$.

This problem is taken from the UKMT Mathematical Challenges.
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