Circles
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Revision notes for CBSE Class 9 Mathematics 'Circles': chords, arcs, segments and sectors, plus the key angle theorems and cyclic quadrilaterals.
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This chapter studies the geometry of a circle through its chords, arcs and angles. You first learn the standard parts of a circle: chord, arc, segment and sector. Then the chapter builds a set of relationships: equal chords subtend equal angles at the centre, the perpendicular from the centre to a chord bisects it, and equal chords are equally far from the centre. These results connect chord lengths, centre distances and angles, so one piece of information determines another. All the definitions are important because exam questions describe figures using this vocabulary.
The central idea is how angles made by the same arc compare. The angle that an arc subtends at the centre is twice the angle it subtends at any point on the remaining circumference. From this follow two well-known results: an angle in a semicircle is a right angle, and angles in the same segment are equal. These are used to find unknown angles from a few given angles, often inside a triangle. A clear diagram and correct identification of the arc involved are the most common sources of mistakes.
The chapter ends with cyclic quadrilaterals, four points on one circle. In such a quadrilateral the sum of each pair of opposite angles is 180°. Since this rule is easy to apply, many angle problems combine it with parallel lines or triangle properties. Practice questions ask you to find one or two unknown angles. Remember also that every theorem has a useful converse, for example two equal chords subtend equal angles at the centre, and if two chords subtend equal angles then the chords are equal.
Key terms
- Circle
- The set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is the centre and the fixed distance is the radius.
- Chord
- A line segment joining any two points on a circle. A diameter is the longest chord of the circle and passes through the centre.
- Arc
- A part of the circumference between two points. The smaller arc is the minor arc and the larger one is the major arc.
- Segment
- The region inside a circle bounded by a chord and one of the two arcs cut off by the chord. The segment formed by the minor arc is the minor segment, and the one formed by the major arc is the major segment.
- Sector
- The region bounded by two radii and the arc included between them. A sector is minor or major according to whether its arc is minor or major.
- Angle subtended by a chord or arc
- If A and B are endpoints of a chord or arc and P is a point on the circle, the angle formed by PA and PB, ∠APB, is the angle subtended by AB at P. The angle subtended at the centre by chord AB is ∠AOB, where O is the centre.
- Distance of a chord from the centre
- The perpendicular distance from the centre to the chord. Equal chords are equidistant from the centre, and in the same circle a chord nearer the centre is longer.
- Cyclic quadrilateral
- A quadrilateral whose four vertices all lie on the same circle. Its opposite angles are supplementary, meaning their sum is 180°.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Perpendicular from centre to chord | OM ⊥ AB ⇒ AM = MB | O is the centre and M is the point where the perpendicular from O meets chord AB. The perpendicular from the centre bisects the chord, so M is the midpoint of AB. |
| Equal chords subtend equal angles at the centre | AB = CD ⇒ ∠AOB = ∠COD | O is the centre. The equal chords are AB and CD, and ∠AOB and ∠COD are the angles they subtend at the centre. |
| Angle at centre vs angle at circumference | ∠AOB = 2∠ACB | O is the centre. A and B are endpoints of an arc that does not contain C, and C is any point on the remaining part of the circle. |
| Angle in a semicircle | AB is a diameter ⇒ ∠ACB = 90° | C is any point on the circle. Since a diameter cuts the circle into two semicircles, the arc AB is a semicircle and its central angle is 180°. |
| Angles in the same segment | ∠APB = ∠AQB | P and Q are points on the same segment cut off by chord AB. Both angles stand on chord AB and are therefore equal. |
| Opposite angles of a cyclic quadrilateral | ∠A + ∠C = 180°, ∠B + ∠D = 180° | ABCD is a cyclic quadrilateral with vertices named in order. Opposite pairs are (A, C) and (B, D). |
Practice questions with answers
1. A chord of a circle is at a distance of 16 cm from the centre. If the radius of the circle is 20 cm, find the length of the chord.
Let O be the centre and M be the foot of the perpendicular from O to chord AB. Since the perpendicular from the centre bisects a chord, AM = half of AB. In right triangle OAM, AM = √(20² − 16²) = √(400 − 256) = √144 = 12 cm. Hence AB = 24 cm.
2. Two equal chords AB and CD of a circle with centre O are given. If ∠AOB = 48°, find ∠COD.
Equal chords subtend equal angles at the centre. Therefore ∠COD = ∠AOB = 48°.
3. In a circle with centre O, ∠AOB = 130°. P is a point on the major arc AB. Find ∠APB.
The angle subtended by arc AB at the centre is twice the angle subtended by the same arc at any point on the remaining circumference. Since P lies on the major arc, the relevant arc AB is the minor arc, whose central angle is 130°, so ∠APB = 130°/2 = 65°.
4. AB is a diameter of a circle and C is a point on the circle. If ∠ABC = 28°, find ∠CAB.
Since AB is a diameter, the angle in the semicircle ∠ACB = 90°. In triangle ABC, ∠CAB = 180° − 90° − 28° = 62°.
5. PQRS is a cyclic quadrilateral. If ∠P = 65° and ∠Q = 80°, find ∠R and ∠S.
Opposite angles of a cyclic quadrilateral are supplementary. Thus ∠R = 180° − 65° = 115°, and ∠S = 180° − 80° = 100°.
6. The perpendicular from the centre of a circle to a chord of length 24 cm is 5 cm. Find the radius of the circle.
Let O be the centre, M be the midpoint of chord AB and OM = 5 cm. Since the perpendicular from the centre bisects the chord, AM = 12 cm. In right triangle OAM, OA = √(12² + 5²) = √(144 + 25) = √169 = 13 cm. The radius is 13 cm.
7. Two chords AB and CD of a circle are at distances 4 cm and 6 cm from the centre. Which chord is longer, and why?
AB is longer. In the same circle, the chord nearer to the centre has the smaller perpendicular distance from the centre, and a smaller distance from the centre means a longer chord. Hence AB, at 4 cm, is longer than CD, at 6 cm.
8. What is a cyclic quadrilateral? State one property of its angles.
A cyclic quadrilateral is a quadrilateral whose four vertices lie on a circle. Its opposite angles are supplementary: the sum of each pair of opposite angles is 180°.
9. A parallelogram ABCD is cyclic. What kind of quadrilateral must it be? Give a reason.
It must be a rectangle. In a cyclic quadrilateral, opposite angles add to 180°. In a parallelogram, opposite angles are equal. If equal angles add to 180°, each is 90°, so every angle of the parallelogram is 90°. Therefore ABCD is a rectangle.
Studying Circles?
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