Circle Theorems - Perpendicular Bisector
Maths: Edexcel A Level Pure Maths, topic 3.2.4
Circle Theorems
Perpendicular bisector goes through centre
- The perpendicular bisector of a chord always goes through the centre of the circle.
Proof
- Draw the perpendicular from centre, O, to the chord AB.
- Draw a radius to A and another to B to form two triangles.
Proof continued
- Their hypotenuses are the same length (because they are both radii) and they share another edge so the triangles are congruent by the ‘RHS’ rule.
- Therefore AM = BM and so the perpendicular splits the chord exactly in half.
Test yourself on Circle Theorems - Perpendicular Bisector
The blue line is a radius of the circle, the red line is a chord. The blue line splits the red line equally, and the two lines intersect at an angle x. What is x?
The blue line is a radius of the circle, the red line is a chord. The blue line splits the red line equally, and the two lines intersect at an angle x. What is x?
Which two lines are the same length?