Coordinate Geometry Circle Properties

Circle properties

Use of the following circle properties:

• The perpendicular from the centre to a chord bisects the chord;
• The tangent at any point on a circle is perpendicular to the radius at that point;
• The angle subtended by an arc at the centre of a circle is twice the angle subtended by the arc at any point on the circumference;
• The angle in a semicircle is a right angle;
• Angles in the same segment are equal;
• The opposite angles in a cyclic quadrilateral add to 180°;
• The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.

You will notice that we have included these circle theorems twice in the TMUA/ESAT specification. They appear both in the M section and also in the MM section. We have included them twice deliberately. We occasionally ask questions on circle theorems in the TMUA/ESAT and we are aware they are a topic that is often covered [perhaps briefly] in introductory maths classes and then forgotten. We do not want you to encounter a TMUA/ESAT question on circle theorems and then realise you cannot answer it because you have forgotten them. And so, we have included them twice – in the M section because they appear in introductory maths classes – and in the MM section to remind you to review your knowledge of the theorems.

We expect you to know and be able to use all the theorems we list. You should also think about the converse of some of the theorems [see the Notes on Logic and Proof]. Whilst we don't expect you to know the proofs of the theorem, you really should make sure you can prove each theorem and that you have a deep understanding of the proof [i.e., you must not try to learn the proof, rather you need to understand what is going on – what are your assumptions, how does the proof work, what geometry is being used etc., etc.].

There are all sorts of techniques you can use when tackling question that link to circle theorems. Here is a list of some techniques you should be comfortable using:

1. Angle chasing – filling in all the angles you can using the theorems above and by looking for isosceles triangles [often made up of two radii] or right angles triangles [in the semicircle].
2. Rotating the diagram – this can often help you get insights into the question.
3. Adding lines – sometimes adding a tangent or diameter or some other line [e.g., a chord] helps you to find the solution.
4. Using dynamic methods – learning to move points around on your diagram in a way that does not affect the solution but makes the question easier to solve.

We would recommend you find a site online that helps you play around with circle theorems. We do not endorse any sites [so this is not an endorsement!] but we found Circle theorems very useful, especially for dynamic methods.