How to do Circle Theorems | 9 Theorems Explained
Introduction
When it comes to the GCSE Maths Higher Paper you will cover a topic known as circle theorems. This is essentially finding missing angles within a circle using a specific set of rules.
However, the questions can also require you to use your knowledge regarding angles in parallel lines as well as triangles and quadrilaterals.
The specific circle theorems that you need to be aware of are as follows:
Theorem 1 – the angle at the centre of a circle is twice the angle at the circumference that is subtended by the same arc
Theorem 2 – every angle at the circumference of a semicircle that is subtended by the diameter of the semi circle is a right angle
Theorem 3 – angles that are subtended at the circumference in the same segment of a circle are equal
Theorem 4 – the sum of the opposite angles of a cyclic quadrilateral is 180°
Theorem 5 – a tangent to the circle is perpendicular to the radius
Theorem 6 – tangents to a circle from an external point to the point of contact are equal
Theorem 7 – the line joining an external point to the centre of the circle bisects the angle between the tangents
Theorem 8 – a radius bisects a chord at 90°
Theorem 9 – the angle between a tangent and a chord through the point of contact is equal to the angle in the other segment
Circle Theorems - Example 1
Take a look at the following question:
The size of angle ACD is also 54° and this is because the angles are in the same segment
Circle Theorems - Example 2
Take a look at the following question:
With this question you can calculate the angle ABC which will be 168° ÷ 2 = 84° because angles at the centre of a circle are twice those at the circumference.
In order to then find angle ADC you can use the fact that opposite angles in a cyclic quadrilateral have a sum of 180°. So angle ADC = 180 – 84 = 96°.
Circle Theorems – Example 3
Take a look at the following question:
Here angle CBO and CAO are both right angles because the tangents meet the radius.
So angle BOC is 180 – 90 – 34 = 56° (angles in a triangle add up to 180°) and this also applies to angle AOC
In order to find angle DOA you can use the fact that angle on a straight line = 180°. So angle DOA = 180 – 56 – 56 = 68°.
Circle Theorems – Question Practice
Try the following questions on your own before looking at the solution:
Question Practice Solution
In this question both angles PAO and PBO are 90° as the tangents meet the radius.
The shape BOAP is a cyclic quadrilateral so angle BOA = 360 – 90 – 90 – 86 = 94°.
Now triangle BOA is an isosceles triangle because the radius is the same so angle OAB = OBA = x.
- 180 = 94 + 2x
- 86 = 2x
- x = 43°
This is of course an overview of the topic and when you are doing your GCSE Maths Circle Theorems revision you could encounter more complex and challenging questions. Whilst this may be the case you need to be applying all the knowledge you have regarding circle theorems, angles and where required algebra.
The topic of Circle theorems does have a lot of rules to try to remember and you need to be asking yourself if it is something that you want to learn. There are actually easier questions to answer without having to learn a lot of theorems. When doing exams, you need to pick topics that you can definitely answer, as there are some circle theorems that can be very complicated.