9. Circles

Exercise 9.1

1. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
2. Show that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

9.2 Perpendicular from the Centre to a Chord

Theorem 9.3

The perpendicular from the centre of a circle to a chord bisects the chord.

Theorem 9.4

The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

9.3 Equal Chords and Their Distances from the Centre

Theorem 9.5

Equal chords of a circle (or congruent circles) are equidistant from the centre.

Theorem 9.6

Chords equidistant from the centre of a circle are equal in length.

9.4 Angle Subtended by an Arc of a Circle

Theorem 9.7

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Theorem 9.8

Angles in the same segment of a circle are equal.

9.5 Cyclic Quadrilaterals

Theorem 9.9

If a line segment joining two points subtends equal angles at two other points on the same side, then the four points lie on a circle.

Theorem 9.10

The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.

Theorem 9.11

If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

9.6 Summary