### 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?

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 16Challenge 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)?