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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?

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Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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.