12. Delia is joining three vertices of a square to make four right-angled triangles.
She can create four triangles doing this, as shown.

How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides?
12. 144 The regular 18 -gon has a circumcircle, that is, a circle passing through all of its vertices. This is also the circumcircle of each right-angled triangle formed. In order for one of these triangle's angles to be a right angle, the opposite side needs to be a diameter, There are 9 possible choices of diameter. For each choice of diameter, there are 8 vertices on each side for the right angle, making 16 choices overall. For each choice of diameter there are 16 choices for the third vertex of the right-angled triangle.