### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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?

### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

# Walk the Plank

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

The figure on the right shows the top left-hand corner of the complete diagram. Note the symmetry which leads to the three measurements of $\frac{1}{2}$. Thus the diagonal of the square can be divided into three portions of lengths:

$\frac{1}{2}$, $x$ and $\frac{1}{2}$ respectively.

The length of the diagonal $= \sqrt{10^2 + 10^2} = \sqrt{200} = 10 \sqrt{2}$.

So $x$ = $10\sqrt{2} - 1$.

This problem is taken from the UKMT Mathematical Challenges.