CBSE Class 9 Maths Chapter 10 Circles Notes

Key Concepts & Must Remember Theorems

• Equal chords of a circle subtend equal angles at the centre.
• If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
• The perpendicular from the centre of a circle to a chord bisects the chord.
• The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
• There is one and only one circle passing through three non-collinear points.
• Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
• Chords equidistant from the centre of a circle are equal in length.
• The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
• Angles in the same segment of a circle are equal.
• The angle in a semicircle is a right angle (90°).

Essential Terminology

Circle

A collection of all points in a plane which are at a fixed distance from a fixed point in the plane.

Radius

The fixed distance from the centre to any point on the circle. (Plural: Radii)

Diameter

A chord passing through the centre of the circle. It is the longest chord and is twice the radius (d = 2r).

Chord

A line segment joining any two points on a circle.

Arc

A continuous piece of a circle. It can be a minor arc (smaller than a semicircle) or a major arc (larger than a semicircle).

Segment

The region between a chord and its corresponding arc. It can be a minor segment or a major segment.

Sector

The region between two radii and their corresponding arc. It can be a minor sector or a major sector.

Secant

A line that intersects a circle at two distinct points.

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on a circle. The sum of opposite angles of a cyclic quadrilateral is 180°.

Understanding Angles Subtended by Arcs

One of the most fundamental and frequently tested concepts in the chapter on Circles is the relationship between the angle subtended by an arc at the centre and the angle subtended by the same arc at any point on the remaining part of the circle. This theorem states that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Let's break this down.

Imagine a circle with centre 'O'. Consider an arc PQ. This arc can be a minor arc or a major arc. Now, consider two angles related to this arc:

1. ∠POQ: This is the angle formed at the centre 'O' by connecting the endpoints of the arc (P and Q) to the centre.
2. ∠PAQ: This is the angle formed at any point 'A' on the remaining part of the circle (the part of the circumference not including arc PQ itself) by connecting A to the endpoints P and Q.

The theorem asserts that ∠POQ = 2 × ∠PAQ. This relationship holds true regardless of where point 'A' is located on the remaining part of the circle. This implies that all angles subtended by the same arc in the same segment are equal. For example, if you pick another point 'B' on the remaining part of the circle, then ∠PBQ would also be equal to ∠PAQ. This is a powerful result often used in proving other properties and solving problems.

A special case arises when the arc is a semicircle. If PQ is a diameter, then arc PQ is a semicircle. The angle subtended by the diameter at the centre is 180° (a straight angle). According to the theorem, the angle subtended by this semicircle at any point on the remaining part of the circle would be half of 180°, which is 90°. This gives us the important conclusion that the angle in a semicircle is a right angle.

Mastering this theorem and its implications is key to solving a wide range of problems involving angles in circles, including those related to cyclic quadrilaterals where opposite angles sum to 180°.

Comparison of Key Theorems

Worked Examples

• Example 1: Angle at Centre Q: In a circle with centre O, if ∠AOB = 80°, where A and B are points on the circle, find ∠ACB where C is a point on the remaining part of the circle. A: According to the theorem, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Therefore, ∠AOB = 2 ∠ACB. Given ∠AOB = 80°, so 80° = 2 ∠ACB. Hence, ∠ACB = 80°/2 = 40°.
• Example 2: Perpendicular to Chord Q: A chord of length 30 cm is drawn in a circle of radius 17 cm. Find the distance of the chord from the centre. A: Let the chord be AB = 30 cm. Radius OA = 17 cm. The perpendicular from the centre O to the chord AB bisects it. Let M be the midpoint of AB. So, AM = MB = 30/2 = 15 cm. In right-angled triangle OMA, by Pythagoras theorem, OM² + AM² = OA². OM² + 15² = 17². OM² + 225 = 289. OM² = 289 - 225 = 64. OM = √64 = 8 cm. The distance of the chord from the centre is 8 cm.

Exam Tip: Mastering Circle Problems

To excel in circle geometry problems, always start by drawing a clear diagram and marking all the given information. Use different colors for radii, chords, and tangents if it helps differentiate. Many problems require combining two or more theorems; for instance, a question might involve both the angle at the centre theorem and properties of cyclic quadrilaterals. Don't forget to state the theorem used for each step in your solution, as this often carries marks. Pay close attention to distinguishing between segments and sectors, and between major and minor arcs. When dealing with cyclic quadrilaterals, remember that the sum of opposite angles is 180° – this is a frequent trap for students who might confuse it with a general quadrilateral.

Quick Revision Check

• Q: What is the measure of the angle in a semicircle? A: The angle in a semicircle is always 90° (a right angle).
• Q: If a chord subtends an angle of 70° at the centre of a circle, what angle will it subtend at any point on the major arc? A: It will subtend an angle of 70°/2 = 35° on the major arc, as the angle at the centre is double the angle at the circumference.
• Q: What is a cyclic quadrilateral? State one important property. A: A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle. An important property is that the sum of any pair of opposite angles is 180°.
• Q: A line from the centre of a circle bisects a chord. What is the relationship between this line and the chord? A: The line is perpendicular to the chord.

Frequently Asked Questions

Frequently Asked Questions