CBSE Class 9 Mathematics Circles Notes Set 07

Circle and Related Terms

A circle is the set of all points in a plane at a fixed distance (radius) from a fixed point (centre). A chord joins two points on the circle; the chord through the centre is a diameter, the longest chord, equal to twice the radius. An arc is a part of the circle; the longer one is the major arc and the shorter the minor arc. A sector lies between an arc and two radii, and a segment lies between an arc and its chord. Circles with the same centre are concentric.

Chords and Centre

Equal chords of a circle subtend equal angles at the centre, and conversely. The perpendicular from the centre to a chord bisects the chord, and the line from the centre to the midpoint of a chord is perpendicular to it. One and only one circle passes through three non-collinear points. Equal chords are equidistant from the centre, and chords equidistant from the centre are equal. If a chord is of length 2c and at distance d from the centre, then r² = c² + d².

Angle Subtended by an Arc

The angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle. Angles in the same segment are equal. An angle in a semicircle is a right angle. If a line segment joining two points subtends equal angles at two other points on the same side of it, the four points lie on a circle.

Cyclic Quadrilaterals

A quadrilateral whose four vertices lie on a circle is cyclic. Its opposite angles are supplementary, and an exterior angle equals the interior opposite angle. Conversely, if a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic. A cyclic parallelogram is a rectangle, and a cyclic trapezium is isosceles.

Locus

The locus of the centre of a circle of given radius r passing through a fixed point is a circle of radius r around that point. The locus of the centres of two equal wheels rolling on a straight road is two lines parallel to the road.

Solved Example 1

PQ and RS are parallel chords of a circle of radius 10 cm, PQ = 16 cm and RS = 12 cm. Find the distance between them.
The distances from the centre are √(100 − 64) = 6 cm and √(100 − 36) = 8 cm. On the same side of the centre the distance is 8 − 6 = 2 cm, and on opposite sides it is 8 + 6 = 14 cm.

Solved Example 2

ABCD is a parallelogram. A circle through A, B, C meets CD (produced if necessary) at E. Prove AD = AE.
ABCE is cyclic, so ∠AEC + ∠ABC = 180°. Also ∠ADE + ∠ADC = 180° and ∠ABC = ∠ADC (opposite angles of a parallelogram). So ∠AEC = ∠ADE, and in △ADE the sides opposite equal angles are equal: AD = AE.