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Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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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

Square Bisection

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Any line which passes though the centre of the square divides the square into two congruent shapes. An example is shown below.

Figure 1

There are an infinite number of suitable lines (lines passing through the centre, at any angle) so there are infinitely many ways the square can be cut in half with a single straight cut.

This problem is taken from the UKMT Mathematical Challenges.
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