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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

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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?

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Two Ladders

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?

Sitting Pretty

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A circle 'sits' on a right-angled triangle, touches two sides of the triangle and has its centre on the hypotenuse of the triangle. The circle has radius $r$ and the triangle has sides of length $x$ and $y$.
Sitting PrettyShow that

$\frac{1}{r} = \frac{1}{x} + \frac{1}{y}.$