Circle theorems

Last time

Before we start: last time, in Angles in polygons and symmetry, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Eight rules, each with a name

Circle theorems look like a lot to learn. But there are only eight, and each has a short name and a picture. In the exam you must name the theorem you use, because the reason is worth a mark of its own. Let us learn the eight with a worked example for each.

The words

A radius goes from the centre to the edge. A chord joins two points on the edge. A diameter is the longest chord. A tangent touches the circle at one point only. An arc is part of the edge. A segment is the region between a chord and an arc. A cyclic quadrilateral has all four corners on the circle.

Centre and circumference

Theorem one: the angle at the centre is twice the angle at the circumference, standing on the same arc. If the angle at the circumference is thirty-five degrees, the angle at the centre is seventy. Theorem two: the angle in a semicircle is ninety degrees. This is the same theorem, with a straight diameter.