Circle Theorems - Semicircle

Circle Theorems - Semicircle

GCSE(H),

The angle at the centre of a circle is twice the angle at the circumference when subtended by the same arc: consider a case where the arc covers 180º.

The angle at the centre of the circle is 180º. The angle at the circumference is therefore 90º.

Examples

1. Points A, B and C are on the circumference of a circle with centre O. Angle ∠ABC has a value of 19º. What is the value of the angle ∠ACB?

Answer: 71º

For a semicircle, the angle at a circumference is 90º. The angles in a triangle add up to 180º. Therefore 180 - 90 - 19 = 71º.

2. What is the value of x in the diagram below?

Answer: 18º

For a semicircle, the angle at a circumference is 90º.

180 = 90 + x + 5x

180 = 90 + 5x

90 = 5x

x = 18º