CBSE Class 9 Mathematics Circles Notes Set 04

Chords and Their Properties

A chord is a line segment joining two points of a circle; the longest chord is a diameter.
1. 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.
2. The perpendicular bisector of a chord passes through the centre.
3. Equal chords are equidistant from the centre, and chords equidistant from the centre are equal. The longer the chord, the nearer it is to the centre.
4. One and only one circle passes through three non-collinear points.
5. A chord of length 2l at distance d from the centre of a circle of radius r satisfies \( r^2 = l^2 + d^2 \).

Angles in a Circle

1. The angle that an arc subtends at the centre is twice the angle it subtends at any point on the remaining part of the circle.
2. Angles in the same segment of a circle are equal.
3. The angle in a semicircle is a right angle; the angle in a major segment is acute and in a minor segment is obtuse.
4. Opposite angles of a cyclic quadrilateral are supplementary, so an exterior angle equals the interior opposite angle.
5. If a line segment subtends equal angles at two points on the same side of it, the four points lie on a circle.

Intersecting Chords and Secants

1. If two chords AB and CD intersect at P (inside or outside the circle), then \( PA \cdot PB = PC \cdot PD \).
2. If PAB is a secant and PT a tangent from the external point P, then \( PT^2 = PA \cdot PB \).
3. The angle between two chords that meet inside a circle is half the sum of the angles that the intercepted arcs subtend at the centre.

Tangents

1. A tangent touches a circle at exactly one point and is perpendicular to the radius at the point of contact.
2. The two tangents from an external point are equal in length and subtend equal angles at the centre; the angle between them is supplementary to the angle subtended by the chord of contact at the centre.
3. The length of a tangent from a point at distance d from the centre of a circle of radius r is \( \sqrt{d^2 - r^2} \).
4. The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
5. If two circles touch externally the distance between the centres is \( r_1 + r_2 \); if they touch internally it is \( r_1 - r_2 \).
6. A quadrilateral circumscribed about a circle has AB + CD = AD + BC.

Worked Example

Example. A circular arch has a span of 30 m and a height of 5 m. Find the radius.
Half the span is 15 m. In a circle a chord of length 30 m at distance (r - 5) from the centre satisfies \( r^2 = 15^2 + (r - 5)^2 \). So \( 0 = 225 - 10r + 25 \) and \( r = 25 \) m.