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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

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

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Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Nicely Similar

Age 14 to 16 Challenge Level:

In the diagram below, a right-angled triangle has a line descending from its right angle, perpendicular to the hypotenuse.

diagram with similar right-angled triangles


If the hypotenuse length is 100cm, and if the line splits the base into 36cm and 64cm parts, what are the side lengths of the original right-angled triangle?

For another way to see this arrangement of right-angled triangles you may like to look at the problem called a Matter of Scale .