Circles Class 9: notes and important questions

Circles: revision notes

1. Circle, chord, arc

A circle is the set of all points in a plane at a fixed distance (the radius \(r\)) from a fixed point (the centre \(O\)). A chord joins two points of the circle; a diameter is a chord through the centre (the longest chord). A chord divides the circle into a minor and a major arc, and the region into two segments.

2. Chords and the centre

• Equal chords subtend equal angles at the centre, and chords that subtend equal angles at the centre are equal.
• The perpendicular from the centre to a chord bisects the chord; the line from the centre to the mid-point of a chord is perpendicular to it.
• The perpendicular bisector of every chord passes through the centre, so there is exactly one circle through three non-collinear points (its centre is where the perpendicular bisectors meet: the circumcentre).

3. Distance of a chord from the centre

If a chord of length \(2l\) is at distance \(d\) from the centre, then \(r^2 = l^2 + d^2\). Equal chords are equidistant from the centre, and chords equidistant from the centre are equal. The nearer a chord is to the centre, the longer it is.

4. Angles subtended by an arc

• The angle an arc subtends at the centre is double the angle it subtends at any point on the remaining part of the circle.
• Angles in the same segment are equal.
• The angle in a semicircle is a right angle.

5. Cyclic quadrilaterals

Points are concyclic if one circle passes through all of them. A quadrilateral with all four vertices on a circle is cyclic: each pair of opposite angles adds up to \(180^\circ\), and conversely a quadrilateral whose opposite angles are supplementary is cyclic. An exterior angle of a cyclic quadrilateral equals the interior opposite angle.

Worked example 1

A chord of a circle of radius \(17\) cm is \(30\) cm long. How far is it from the centre?

Half-chord \(15\), so \(d = \sqrt{17^2 - 15^2} = \sqrt{64} = 8\) cm.

Worked example 2

\(O\) is the centre and \(\angle AOB = 130^\circ\). Find \(\angle ACB\) for a point \(C\) on the major arc.

\(\angle ACB = \dfrac12 \times 130^\circ = 65^\circ\).

Worked example 3

In cyclic quadrilateral \(PQRS\), \(\angle P = 3\angle R\). Find \(\angle P\).

\(\angle P + \angle R = 180^\circ\), so \(4\angle R = 180^\circ\), \(\angle R = 45^\circ\), \(\angle P = 135^\circ\).

Common errors

• Using the whole chord instead of half of it in \(r^2 = l^2 + d^2\).
• Halving the angle at the centre when the point is on the same arc (the minor arc) as the chord: then the angle is \(180^\circ\) minus half.
• Adding adjacent angles of a cyclic quadrilateral to \(180^\circ\): only opposite angles are supplementary.
• Two parallel chords: forgetting to check whether they are on the same side of the centre or on opposite sides.

Exam tips

• Mark the centre and draw the perpendicular to the chord: most chord questions become a right triangle.
• In proofs, give the reason for each step in brackets: "(angles in the same segment)", "(opposite angles of a cyclic quadrilateral)".