### 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?

### 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

### Pinned Squares

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

# Folding Fractions

##### Age 14 to 16 Challenge Level:

You may wish to look at Folding squares before trying this problem.

 You can divide the long diagonal of a square into different fractions by folding. In the first image the second fold joins a corner of the square to the midpoint of the opposite side. In the second image the second fold joins a corner of the square to a point $\frac{1}{8}$ of the way along the opposite side. This problem is about the fractions of the long diagonal of a square which you can construct in this way. To start with, we shall only consider points on the side of the square which can easily be found by folding. That is, $\frac{1}{2}$s or $\frac{1}{4}$s or $\frac{1}{8}$s and so on.
Investigate the fractions of the long diagonal of a square that can be created in the way described above. Here are some examples to think about:

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Can you extend the findings and make generalisations?
Can you justify your generalisations?

What about starting with fractions of the side of the square that are not so easily found by folding?