### 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?

# Something in Common

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

This idea uses the work on "tilted squares" in September 2004 but with a twist.

Vector dot product is useful but certainly not essential..

Extending the idea to what squares are and are not possible gives lots of opportunities for students who get through the 2D version very easily but at the same time develops a feel for 3D coordinate patterns.