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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

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)?