Grade eight, Edexcel GCSE Maths

Revision Notes

What are circle theorems?

You would have learned a lot of angle facts for your GCSE – angles in polygons, angles with parallel lines.  Circle Theorems deal with angle facts that occur with shapes and lines drawn within and connected to a circle.  There are quite a few of them!

What do I need to know?

Firstly, it helps to be familiar with the names of parts of a circle and lines within and outside of a circle as per the diagram below:

Parts of a circle, IGCSE & GCSE Maths revision notes

You need to know the following circle theorems:

1.The angle at the circumference in a semicircle is a right angle – see below

Right angle in a semicicrcle, IGCSE & GCSE Maths revision notes

2. Angles at the circumference subtended by the same arc are equal – see below

Equal subtend arc, IGCSE & GCSE Maths revision notes

3. Angle subtended by an arc at the centre is twice the angle at the circumference – see below

Centre twice circumference, IGCSE & GCSE Maths revision notes

4. A radius and a tangent are perpendicular

Radius tangent 90, IGCSE & GCSE Maths revision notes

5. Two tangents that meet are equal in length 

Equal length tangents, IGCSE & GCSE Maths revision notes

6. Opposite angles in a cyclic quadrilateral add up to 180° 

Cyclic quadrilaterals, IGCSE & GCSE Maths revision notes

7. The perpendicular bisector of a chord is a radius

Radius chord bisect, IGCSE & GCSE Maths revision notes

8. Alternate Segment Theorem 

Alternate Segment Theorem, IGCSE & GCSE Maths revision notes
These are not in any particular order but generally get more difficult to work with!

To solve harder problems you may need to use the angle facts you are already familiar with from triangles, polygons, parallel lines and you may have to use circumference and area so ensure you’re familiar with the two formulas: C=πd and A=πr2.

The first three angles theorems …

1.The angle at the circumference in a semicircle is a right angle

Right angle in a semicicrcle, IGCSE & GCSE Maths revision notes

This theorem should be self-explanatory form its name/title.  The semicircle arises if you ignore the right-hand side of the diameter in the diagram above.  Look out for triangles hidden amongst other lines/shapes within the circle

2. Angles at the circumference subtended by the same arc are equal

Equal subtend arc, IGCSE & GCSE Maths revision notes

This is easier to see that its name!  Subtended means the equal angles are created by drawing chords from the ends of the arc PQ. Theses chords may or may not pass through the centre. 

3. Angle subtended by an arc at the centre is twice the angle at the circumference

Centre twice circumference, IGCSE & GCSE Maths revision notes

Similar to above the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the arc PQ.

This theorem can also happen when the ‘triangle parts’ overlap:

Triangle overlap, IGCSE & GCSE Maths revision notes

Example

Circle Theorems 1 Example 1, IGCSE & GCSE Maths revision notes
Q1-Circle-Theorems-1, IGCSE & GCSE Maths revision notes
Circle Theorems 1 Example 2, IGCSE & GCSE Maths revision notes
Q1-Circle-Theorems-2, IGCSE & GCSE Maths revision notes
Circle Theorems 1 Example 3, IGCSE & GCSE Maths revision notes
Q2-Circle-Theorems-1, IGCSE & GCSE Maths revision notes
Circle Theorems 1 Example 4, IGCSE & GCSE Maths revision notes
Q2-Circle-Theorems-2, IGCSE & GCSE Maths revision notes
Circle Theorems 1 Example 5, IGCSE & GCSE Maths revision notes

Question

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Edexcel GCSE Maths Notes

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After 24 years teaching A Level Further Maths, Maths whizz Simon focused his attention on tutoring. He’s helped thousands of young people make sense of Maths and now he’s helping you too – aren’t you lucky! He’s even worked as an Edexcel examiner, so knows exactly what they’re looking for.