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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?

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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.