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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?

Triangular Teaser

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

The diagram below shows isosceles triangles $T$ and $U$. The perpendicular from the top vertex to the base divides an isosceles triangle into two congruent right-angled triangles as shown in both $T$ and $U$. Evidently, by Pythagoras' Theorem, $h = 4$ and $k = 3$. So both triangles $T$ and $U$ consist of two $3$, $4$, $5$ triangles and therefore have equal areas.

This problem is taken from the UKMT Mathematical Challenges.
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