Circle Theorems

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

1. How circle questions work

Many circle questions ask for an angle and the theorem that justifies it, with a mark for each
On some questions the reason is a whole part on its own, so the wording has to be right
Where the answer takes two steps, two reasons are expected: give one for each step, not just the last one
The ordinary angle facts stay available throughout: angles in a triangle, angles on a line, angles at a point, isosceles triangles and angles in a quadrilateral
Mark the radii on a diagram first, since any two of them make an isosceles triangle and that is often the missing step
Diagrams are not to scale, so never measure

Two angle theorems are examined: the angle in a semicircle is 90°, and the angle between a tangent and a radius is 90°
The ordinary angle facts above do the rest of the work, so a question is usually a short chain of those with one circle theorem in the middle

Where a triangle has the diameter as one side and its third corner on the circumference, the angle at that corner is a right angle
To spot it, check that one side runs through the centre and that every corner lies on the circumference
Whichever corner faces the diameter is the one holding the right angle

Worked example

Using the semicircle and the triangle

The diameter LN is drawn in a circle, and M is a third point on the circumference. Angle MLN is 28°. Find angle MNL, giving reasons.

Solution:

A tangent touches the circle at exactly one point, and the radius drawn to that point is perpendicular to it
Draw the radius in yourself whenever a tangent appears, because the right angle is usually what unlocks the question
A mark scheme has accepted "tangent and radius are perpendicular" as well as the ninety-degree form, and allowed diameter in place of radius