### Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

### Look Before You Leap

The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?

# Folding Fractions

##### Stage: 4 Challenge Level:

Richard of Mearns Castle High School sent us in this solution:

Notice that $AE = \frac{1}{n} AB = \frac{1}{n}x$

We can say that $\triangle AEF$ and $\triangle DCF$ are similar. This is because $\angle AFE$ is equal to $\angle DFC$ (vertically opposite). Also $\angle FAE = \angle FCD$ and $\angle AEF = \angle CDF$ (alternate angles).

Since the triangles are similar we can say that the ratios of corresponding sides are the same. Therefore:
\eqalign{ \frac {DC}{AE} &= \frac{FC}{AF}\cr \frac{x}{\frac{x}{n}}&= \frac{FC}{AF}\cr n &= \frac{FC}{AF} }
Hence$$FC = AF \times n$$
So $DE$ cuts $AC$ at the ratio $1:n$.