### 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.

# Centre Square

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

Let $r$ be the radius of each of the larger circles.
The sides of the square are equal to $r+1$, the sum of the two radii.
The diagonal of the square is $2r$.

By Pythagoras,         $$(r+1)^2+(r+1)^2 = (2r)^2$$Simplifying gives:      $$2(r+1)^2 = 4 r^2$$ i.e.                          $$(r+1)^2 = 2r^2$$
so$$r+1 = \sqrt{2}r$$

[$-\sqrt{2}r$ is not possible since $r+1> 0$].

Therefore $(\sqrt{2} - 1)r = 1$.
Hence $r= \frac {1}{\sqrt{2}-1} = \sqrt{2} + 1$.

This problem is taken from the UKMT Mathematical Challenges.