### Building Tetrahedra

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

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?

### 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.

# Isosceles Reduction

##### Stage: 4 Short Challenge Level:

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.