## Dotty Relationship

On the grid below, join the two blue dots, A and B, together with a straight line.

Now join the green dots, C and D, with a straight line.

At what angle do the two lines cross?

Investigate the number of squares "along" and "down" from A to B compared with the number of squares "along" and "up" from C to D. What do you notice?

Using what you have found out, can you draw lines that are perpendicular (at $90^{\circ}$) to the lines drawn below?

Is there only one solution each time?

Do the two lines have to be the same length? Why or why not?

This problem could be linked to coordinates. It could also form an introduction to vectors at a higher level. It is a powerful geometrical investigation. It has the potential to quickly lead to generalisations that pupils can apply in the second part of the question.

The notion of "how long is a line?" would form an excellent discussion point in the plenary.