Circle Theorems - Chords

Circle Theorems - Chords

GCSE(H),

A line drawn from the centre of a circle to bisect a chord will be perpendicular to the chord.

Examples

1. A circle of radius 10cm has a chord drawn across it of length 12cm. What is the distance from the midpoint of the chord to the centre of the circle?

Answer: 6 cm

The distance from the midpoint of the chord to the circumference is 6cm. This forms a right angled triangle with the radius. Using Pythagoras:

`10^2 = 6^2 + x^2`

`100 - 36 = x^2`

`x^2 = 64`

`x = 6`

2. What is the size of the angle ∠AOB?

Answer: 82º

∠AOM = 180 - 90 - 49 = 41º

∠AOB is twice the size of ∠AOM = 2 x 41 = 82º