Circle Angle Theorems

Grade 6

The six theorems

1. Angle at the centre is twice the angle at the circumference when both are subtended by the same arc.

2. Angles in the same segment are equal — any two angles at the circumference on the same side of a chord are equal.

3. Angle in a semicircle is always 90° — the angle subtended by a diameter at the circumference is a right angle.

4. Opposite angles in a cyclic quadrilateral sum to 180°.

5. Tangent–chord angle equals the angle in the alternate segment (alternate segment theorem).

6. Two tangents from an external point are equal in length.

Using them in questions

In exam questions, always state the theorem name and give the reason. "Angle at the centre = twice angle at circumference" scores the reasoning mark — just writing the number does not.

Worked example

EXAMPLE

O is the centre of a circle. Angle AOB = 124° where A and B are on the circumference. Find the angle ACB where C is another point on the major arc.

Angle at centre = twice angle at circumference (same arc AB). Angle ACB = 124° ÷ 2 = 62°.

Try this one

A chord \( PQ \) is drawn in a circle of centre \( O \) and radius \( 9 \) cm. The perpendicular distance from \( O \) to the chord \( PQ \) is \( 4 \) cm.

Work out the length of the chord \( PQ \). Give your answer to 1 decimal place. [4]

Nova's hint:

The perpendicular from the centre to a chord bisects the chord, creating a right-angled triangle.

Ask yourself: 'If I drop a perpendicular from the centre, what right-angled triangle appears, and which side does Pythagoras give me?' The radius is the hypotenuse.

Try to Solve it with Nova?