Circle Theorems 2
Maths: Edexcel IGCSE Higher, topic 4.5.3
Circle Theorems
Opposite angles in a cyclic quadrilateral
- Opposite angles in a cyclic quadrilateral add up to 180°.
- Here x + y = 180°
Opposite angles add to 180°
Proof
- Label the opposite angles, then draw two radii from the other corners.
- Use ‘angle at centre twice that at the circumference’ to get the angles at the centre as 2x, 2y.
Proof continued
- Since angles at a point add up to 360°:
- 2x + 2y = 360° and so x + y = 180°.
Circle Theorems
Lengths of tangents
- Two tangents to the same circle from the same point (C) will be the same length.
- Here AC = BC
Proof
- Add radii from the points, A and B, where the tangents touch and draw a line from the centre of the circle, O to where the tangents meet, C.
Proof continued
- These are congruent triangles (by using the 'RHS' rule).
- They share a common side (length x) and another side which is a radius.
- Therefore the remaining side, the tangent, is the same length for both.
Test yourself on Circle Theorems 2
What do the opposite angles in a cyclic quadrilateral add up to?
Which of these are true about the diagram?
Which of these are true about the diagram?
Which of these are true about the diagram?