### 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 Squares

##### What fractions of the diagonal do you think your new fold has created?

Measure the two sections of the diagonal and compare their lengths to the diagonal's total length.
Is this what you expected?

Create or draw some more $10$cm squares and repeat the process.
Do you always end up with the same answer?

What fractions does the second fold appear to divide the diagonal into?

Does this appear to be the case for squares of different sizes?
Can you produce a convincing mathematical argument or proof that justifies what you have found?

Would the same work if you started with a rectangle or a parallelogram or a trapezium?