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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?

Folding Squares

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

Start with a square...

Fold to create a diagonal running from bottom left to top right.
Find the midpoint of the base by folding.
Fold the paper again, joining the top left corner to the midpoint of the base.
What fractions of the diagonal do you think your new fold has created?

Measure the two sections of the diagonal and compare their lengths.
What do you notice?
Is this what you expected?
Can you produce a convincing mathematical argument or proof that justifies what you have found?

You might like to try A Parallelogram Trisection and Folding Fractions next.