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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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From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Two Ladders

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

Triangles ABC and CDE are similar and the ratio of DE:AB is 2:1,
so the ratio CN:CM must also be 2:1.
Since NM = DE =10m,
CM = 3 1/3 metres

The ladders cross at a height of 3 1/3 metres.