### 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:

Richard of Mearns Castle High School sent us in this solution:

Notice that $AE = \frac{1}{n} AB = \frac{1}{n}x$

We can say that $\triangle AEF$ and $\triangle DCF$ are similar. This is because $\angle AFE$ is equal to $\angle DFC$ (vertically opposite). Also $\angle FAE = \angle FCD$ and $\angle AEF = \angle CDF$ (alternate angles).

Since the triangles are similar we can say that the ratios of corresponding sides are the same. Therefore:
\eqalign{ \frac {DC}{AE} &= \frac{FC}{AF}\cr \frac{x}{\frac{x}{n}}&= \frac{FC}{AF}\cr n &= \frac{FC}{AF} }
Hence$$FC = AF \times n$$
So $DE$ cuts $AC$ at the ratio $1:n$.