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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?

Squirty

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Using a ruler and compass only it is possible to fit a square into 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:

 

Square inscribed in a triangle

 

How is this possible?

Prove why this works.

It is easy to draw a square ABCD with BC on the base PR of the triangle and the vertex D on the side PQ

 

Small square drawn in left-hand corner of the triangle

 

What happens to the point A as you enlarge the square? The interactivity may help.

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