CBSE Class 9 Mathematics Circles Notes Set 06

Circle Terms

A circle is the set of points at a fixed distance (radius) from a fixed point (centre). Chord: a segment joining two points on the circle; diameter: the longest chord, through the centre; circumference: the length of the circle, 2πr.
A diameter splits the circle into two semicircles. An arc smaller than a semicircle is a minor arc and a larger one is a major arc; the degree measure of an arc equals the angle it subtends at the centre.
Length of an arc = (angle ÷ 360) × 2πr. Area of a sector = (angle ÷ 360) × πr². A segment is the region between a chord and its arc: area = area of sector minus area of the triangle. A quadrant is a quarter of a circle. Congruent circles have equal radii.

Chords and the Centre

Equal chords subtend equal angles at the centre, and conversely. Equal chords cut off congruent arcs, and conversely.
The perpendicular from the centre to a chord bisects the chord, and the line from the centre to the mid-point of a chord is perpendicular to it.
Equal chords are equidistant from the centre, and chords equidistant from the centre are equal. Of two chords, the longer one is nearer to the centre.
Exactly one circle passes through three non-collinear points; no circle passes through three collinear points; infinitely many circles pass through two given points.

Angles in a Circle

The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining part of the circle.
The angle in a semicircle is a right angle, and conversely an arc subtending a right angle at a point of the circle is a semicircle.
Angles in the same segment are equal. If a segment subtends equal angles at two points on the same side, the four points are concyclic.

Cyclic Quadrilaterals

A quadrilateral whose vertices lie on a circle is cyclic. Its opposite angles add up to 180°, and conversely a quadrilateral with a pair of supplementary opposite angles is cyclic.
If a side of a cyclic quadrilateral is produced, the exterior angle equals the interior opposite angle.
A cyclic parallelogram is a rectangle; a cyclic trapezium is isosceles; the angle bisectors of a cyclic quadrilateral form another cyclic quadrilateral.

Tangents and Secants

A tangent touches the circle at exactly one point and is perpendicular to the radius at that point. A secant meets the circle at two points.
Tangents from an external point are equal in length, subtend equal angles at the centre and are equally inclined to the line joining the point to the centre.
Tangent-chord angle: the angle between a tangent and a chord equals the angle in the alternate segment.
If a secant PAB and a tangent PT are drawn from P, then PA × PB = PT²; for two secants PAB and PCD, PA × PB = PC × PD.

Common Tangents of Two Circles

Let the radii be r1 and r2 and the distance between centres d.
One circle inside the other (d < r1 − r2): no common tangent. Internal touching (d = r1 − r2): 1 common tangent. Intersecting (r1 − r2 < d < r1 + r2): 2 common tangents. External touching (d = r1 + r2): 3 common tangents. Separate circles (d > r1 + r2): 4 common tangents.

Worked Examples

Example 1: Parallel chords AB = 8 cm and CD = 6 cm lie on opposite sides of the centre of a circle of radius 5 cm. Distances from the centre are √(25 − 16) = 3 cm and √(25 − 9) = 4 cm, so the chords are 7 cm apart.
Example 2: Two equal chords AB and AC of a circle: the centre lies on the bisector of ∠BAC. Triangles BAM and CAM (M on BC) are congruent by SAS, so AM is the perpendicular bisector of BC and passes through the centre.
Example 3: BD and CA meet at X with ∠DXC = 40° and ∠XCD = 25°. Then ∠XDC = 115° = ∠BAC (same segment), and the reflex angle BOC = 2 × 115° = 230°.
Example 4: A pair of opposite sides of a cyclic quadrilateral are equal. Then the angles they subtend at the circle are equal, which gives equal angles ∠ADC and ∠BAD, so the diagonals AC and BD are equal.