Circle
ICSE Class 9 · Mathematics · Chapter 16
What this chapter covers
- The circle and its parts
- The line from the centre that bisects a chord
- The perpendicular from the centre bisects the chord
- Equal chords are equidistant from the centre
- Chords equidistant from the centre are equal
- One and only one circle through three points
- Arcs that subtend equal angles at the centre
- Equal chords cut off equal arcs
Key definitions
Congruent circles are circles with equal radii.
Concentric circles are circles that have the same centre but different radii.
Equal arcs are arcs of the same circle, or of congruent circles, that can be placed one on the other so that they coincide exactly.
Key points
- A circle is the set of points in a plane at a fixed distance (the radius) from a fixed point (the centre); all radii of a circle are equal.
- A chord joins two points of a circle; a diameter is a chord through the centre, of length 2r, and is the longest chord.
- The distance of a chord from the centre is the length of the perpendicular from the centre to the chord.
- Theorem 1 (with proof, SSS): the line from the centre to the mid-point of a chord that is not a diameter is perpendicular to the chord.
- Theorem 2 (without proof): the perpendicular from the centre to a chord bisects the chord.
- The working triangle: r2 = d2 + x2, where x is half the chord and d its distance from the centre; always use half the chord.
- The perpendicular bisector of every chord passes through the centre.
- Theorem 3 (with proof, RHS): equal chords are equidistant from the centre. Theorem 4 (without proof): chords equidistant from the centre are equal.
- The nearer a chord is to the centre, the longer it is.
- Theorem 5 (with proof): one and only one circle passes through three non-collinear points; its centre is where the perpendicular bisectors meet. No circle passes through three collinear points.
- Two distinct circles meet in at most two points.
- Theorem 6 (with proof, by superposition): arcs subtending equal angles at the centre are equal, and equal arcs subtend equal angles at the centre.
- Equal chords subtend equal angles at the centre (SSS), and chords subtending equal angles at the centre are equal (SAS).
- Theorem 7 (without proof): equal chords cut off equal arcs, and chords of equal arcs are equal.
- Every arc and chord property holds only within one circle or within congruent circles.
Questions answered
Define a chord of a circle, and explain why a diameter is the longest chord.
A chord is a line segment whose two end points both lie on the circle. For any chord AB not through the centre O, triangle OAB gives AB < OA + OB = 2r, while a diameter has length exactly 2r; so no chord is longer than a diameter.
State the theorem about the line joining the centre of a circle to the mid-point of a chord. Why does the statement exclude a diameter?
A straight line drawn from the centre of a circle to bisect a chord which is not a diameter is at right angles to the chord. For a diameter the mid-point is the centre itself, so there is no line from the centre to it with a definite direction.
In the proof of Theorem 1, which congruence criterion is used, and how is the right angle obtained from the congruence?
SSS, using OA = OB (radii), AM = BM (mid-point) and OM common. The congruence gives ∠ OMA = ∠ OMB; these are angles on a straight line adding to 180°, so each is 90°.
The radius of a circle is 17 cm and a chord is 8 cm from the centre. Find the length of the chord.
Half-chord = √(172 - 82) = √(289 - 64) = √(225) = 15 cm, so the chord is 30 cm.
A chord 10 cm long is 12 cm from the centre of a circle. Find the diameter of the circle.
r2 = 122 + 52 = 169, so r = 13 cm and the diameter is 26 cm.
Two chords of a circle are 4 cm and 6 cm from the centre. Which chord is longer, and why?
The chord 4 cm from the centre. From x2 = r2 - d2, a smaller distance d gives a larger half-chord x, so the chord nearer the centre is longer.
AB and CD are chords of a circle with centre O, and the perpendiculars from O to them are each 7 cm long. If AB = 9 cm, what is CD? Name the result used.
CD = 9 cm, because chords equidistant from the centre are equal (the converse of Theorem 3, stated without proof in the syllabus).
Two parallel chords of a circle of radius 5 cm are 6 cm and 8 cm long and lie on opposite sides of the centre. Find the distance between them.
Distances from the centre are √(25 - 9) = 4 cm and √(25 - 16) = 3 cm. On opposite sides, the distance between the chords is 4 + 3 = 7 cm.
Past board exam questions
ICSE 2019 sample paper · 4 marks
'O' is the centre of a circle and AB is parallel to CD, the chords AB and CD lying on opposite sides of O. AB = 24 cm and distance between the chords AB and CD is 17 cm. If the radius of the circle is 13 cm, find the length of the chord CD.
Show answer
Draw OM ⊥ AB and ON ⊥ CD; M, O, N are collinear (the chords are on opposite sides of O). The perpendicular from the centre bisects the chord, so AM = 12 cm.
In right △OMA: OM = √(132 - 122) = √(25) = 5 cm.
ON = MN − OM = 17 − 5 = 12 cm.
In right △ONC: CN = √(132 - 122) = 5 cm.
CD = 2 × CN = 10 cm.
ICSE 2019 sample paper · 3 marks
A, B and C are points on a circle with centre O, B lying on the minor arc AC, and chord AC is drawn. Arc AB = twice arc BC and ∠AOB = 80°. Find: (i) ∠BOC (ii) ∠OAC
Show answer
(i) Angles at the centre are proportional to the arcs they subtend. Arc AB = 2 × arc BC, so ∠AOB = 2∠BOC ⇒ ∠BOC = 40°.
(ii) ∠AOC = ∠AOB + ∠BOC = 80° + 40° = 120°.
In △OAC, OA = OC (radii), so ∠OAC = ∠OCA = (180° - 120°)/(2) = 30°.