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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

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Pinned Squares

What is the total number of squares that can be made on a 5 by 5 geoboard?

Something in Common

Age 14 to 16 Challenge Level:

You might also like to look at the problem called "Tilted Squares" published in Sept04.

Some cubes or a construction set might help you to visualise a 3D grid.

How many points are there that are $ \surd 3$ units from (0,0,0)?