### 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

Can you spot a cunning way to work out the missing length?

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?

# Squirty

##### Stage: 4 Challenge Level:

Using a ruler and compass only, it is possible to construct a square in any triangle so that one side of the square rests on one side of the triangle, and the other two vertices of the square touch the other two sides of the triangle:

Can you find a way to construct the square, for any triangle?
Can you explain why your method works?