Circles Class 9 Notes | Geometry Revision

Key Points to Remember

• A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre).
• The diameter is the longest chord of a circle and divides the circle into two equal semicircles.
• A perpendicular from the centre to a chord bisects the chord.
• The perpendicular bisector of a chord always passes through the centre of the circle.
• Equal chords of a circle are equidistant from the centre.
• Chords equidistant from the centre 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°).
• The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Key Definitions for Circles

Circle

The locus of a point that moves in a plane such that its distance from a fixed point (center) remains constant.

Radius

The distance from the center to any point on the circle. All radii of a circle are equal.

Chord

A line segment joining any two points on the circumference of a circle.

Diameter

A chord that passes through the center of the circle. It is the longest chord and is equal to twice the radius (D = 2r).

Arc

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

Sector

The region bounded by two radii and an arc of the circle.

Segment

The region bounded by a chord and an arc of the circle. It can be a minor segment or a major segment.

Secant

A line that intersects a circle at two distinct points.

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on the circumference of a circle.

Properties of Chords and Arcs

Understanding the properties of chords and arcs is fundamental to solving problems related to circles. A chord is a straight line segment whose endpoints both lie on the circle. The diameter is a special type of chord that passes through the center, making it the longest possible chord in any given circle. An important theorem states that the perpendicular from the center of a circle to a chord bisects the chord. This means if you draw a line from the center that is perpendicular to a chord, it will cut the chord into two equal halves. Conversely, the perpendicular bisector of any chord of a circle will always pass through the center of the circle. This property is often used to find the center of a circle if you're given two chords.

Another crucial set of theorems deals with the relationship between equal chords and their distance from the center. It states that equal chords of a circle are equidistant from the center. This implies that if two chords have the same length, their perpendicular distances from the center will be identical. The converse is also true: chords that are equidistant from the center of a circle are equal in length. These theorems are vital for comparing chord lengths or distances, or for proving congruence of triangles formed within the circle. Furthermore, arcs are portions of the circle's circumference. If two arcs of a circle are congruent, then their corresponding chords are equal, and conversely, if two chords are equal, then their corresponding arcs are congruent. These relationships allow us to link linear segments (chords) with curved segments (arcs) within a circle, forming the basis for many geometric proofs and calculations.

Key Angle Properties in a Circle

Worked Examples

• {"heading":"Example 1: Chord Length","bodyMarkdown":"Q: A chord of length 16 cm is drawn in a circle of radius 10 cm. Find the distance of the chord from the centre.\nA: Let the chord be AB = 16 cm, and radius OA = 10 cm. Draw OM perpendicular to AB. Then AM = MB = 16/2 = 8 cm. In right-angled ΔOMA, by Pythagoras theorem: $OM^2 + AM^2 = OA^2 \\implies OM^2 + 8^2 = 10^2 \\implies OM^2 + 64 = 100 \\implies OM^2 = 36 \\implies OM = 6 \\text{ cm}$. The distance of the chord from the centre is 6 cm."}
• {"heading":"Example 2: Angle Subtended","bodyMarkdown":"Q: In a circle with centre O, if ∠ABC = 40°, find ∠AOC, where A, B, C are points on the circle.\nA: 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. Here, arc AC subtends ∠AOC at the centre and ∠ABC at point B on the remaining part. Therefore, ∠AOC = 2 × ∠ABC = 2 × 40° = 80°."}

Exam Tip for Circles Chapter

Always draw clear and labelled diagrams for every problem involving circles. This helps visualize the relationships and apply the correct theorems. For proofs, clearly state the given, what is to be proved, and the construction (if any), then write logical steps, citing the theorems used. Common mistakes include confusing angles subtended at the centre with those at the circumference, or forgetting that the angle in a semicircle is always 90 degrees. Pay special attention to cyclic quadrilaterals; their properties are frequently tested. Practice applying the converse theorems as well, such as 'chords equidistant from the centre are equal'.

Quick Revision Check

• Q1: What is the relationship between the angle subtended by an arc at the center and the angle subtended by the same arc at any point on the remaining part of the circle? A1: The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
• Q2: If two chords of a circle are equal, what can be said about their distances from the center? A2: If two chords of a circle are equal, they are equidistant from the center.
• Q3: What type of angle is formed when a diameter subtends an angle at any point on the circumference? A3: It forms a right angle (90°).
• Q4: What is a cyclic quadrilateral? A4: A quadrilateral whose all four vertices lie on the circumference of a circle.

Frequently Asked Questions