Circle theorems · Year 12
In Year 12 you learn: proving circle theorems.
Circle theorems are facts about the angles formed by chords, radii and tangents of a circle. They come from isosceles triangles (two radii are always equal) and are used, with reasons, to find missing angles.
Key facts
- The angle at the centre is twice the angle at the circumference standing on the same arc.
- The angle in a semicircle is 90°.
- Angles in the same segment (standing on the same chord, same side) are equal.
- Opposite angles of a cyclic quadrilateral add up to 180°.
- A tangent meets the radius at 90°; two tangents from a point are equal in length.
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
Worked examples
Example 1
PA and PB are tangents to the circle with centre O, and angle APB = 50°. Find angle AOB and angle OAB.
Answer: AOB = 130°, OAB = 25°