Class 9 Mathematics Notes Chapter 10 (Circles) – Mathematics Book
Alright class, let's focus on Chapter 10: Circles. This is a crucial chapter for geometry, and understanding its concepts thoroughly is important for your exams. We'll break down the key definitions, theorems, and properties step-by-step.
Chapter 10: Circles - Detailed Notes for Government Exam Preparation (Based on NCERT Class 9)
1. Introduction & Basic Terms
- Circle: A circle is the collection (or locus) of all points in a plane that are at a fixed distance from a fixed point in the same plane.
- The fixed point is called the Centre (O).
- The fixed distance is called the Radius (r).
- Interior & Exterior: The plane is divided into three parts by a circle:
- Inside the circle (Interior): Points whose distance from the centre is less than the radius.
- On the circle: Points whose distance from the centre is equal to the radius.
- Outside the circle (Exterior): Points whose distance from the centre is greater than the radius.
- Chord: A line segment joining any two points on the circle.
- Diameter: A chord that passes through the centre of the circle. It is the longest chord of a circle. (Diameter = 2 × Radius).
- Arc: A continuous piece of a circle between two points.
- Minor Arc: The shorter arc connecting two points.
- Major Arc: The longer arc connecting two points.
- Semicircle: When the two points are the ends of a diameter, both arcs are equal and are called semicircles.
- Circumference: The total length of the boundary of the circle.
- Segment: The region between a chord and either of its arcs.
- Minor Segment: The region bounded by a chord and the minor arc.
- Major Segment: The region bounded by a chord and the major arc.
- Sector: The region between two radii and the arc connecting their endpoints on the circle.
- Minor Sector: The region bounded by two radii and the minor arc.
- Major Sector: The region bounded by two radii and the major arc.
2. Key Theorems and Properties
(Remember: For many government exams, you need to know the statements and applications of these theorems, not necessarily the rigorous proofs)
-
Theorem 10.1: Equal chords of a circle subtend equal angles at the centre.
- Meaning: If chord AB = chord CD in a circle with centre O, then ∠AOB = ∠COD.
-
Theorem 10.2 (Converse of 10.1): If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
- Meaning: If ∠AOB = ∠COD in a circle with centre O, then chord AB = chord CD.
-
Theorem 10.3: The perpendicular from the centre of a circle to a chord bisects the chord.
- Meaning: If OM ⊥ AB (where O is the centre and AB is a chord), then AM = MB.
-
Theorem 10.4 (Converse of 10.3): The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
- Meaning: If O is the centre and M is the midpoint of chord AB (AM = MB), then OM ⊥ AB.
-
Important Note: There is one and only one circle passing through three given non-collinear points.
-
Theorem 10.5: Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
- Meaning: If chord AB = chord CD in a circle with centre O, and OM ⊥ AB, ON ⊥ CD, then OM = ON (where OM and ON are the distances of the chords from the centre).
-
Theorem 10.6 (Converse of 10.5): Chords equidistant from the centre of a circle are equal in length.
- Meaning: If OM ⊥ AB, ON ⊥ CD, and OM = ON in a circle with centre O, then chord AB = chord CD.
-
Theorem 10.8: 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.
- Meaning: Let arc AB subtend ∠AOB at the centre O and ∠ACB at any point C on the remaining part of the circle. Then ∠AOB = 2 × ∠ACB.
- Application: This is very useful for finding unknown angles.
-
Theorem 10.9: Angles in the same segment of a circle are equal.
- Meaning: If C and D are any two points on the same arc AB (major or minor), then ∠ACB = ∠ADB.
- Derivation: Both angles are half the angle subtended by arc AB at the centre.
-
Property: Angle in a semicircle is a right angle (90°).
- Reason: The arc of a semicircle subtends 180° at the centre. Any angle on the circumference subtended by this arc will be half of 180°, which is 90°.
-
Cyclic Quadrilateral: A quadrilateral whose all four vertices lie on a circle.
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Theorem 10.11: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
- Meaning: If ABCD is a cyclic quadrilateral, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.
-
Theorem 10.12 (Converse of 10.11): If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.
- Meaning: If in quadrilateral ABCD, ∠A + ∠C = 180° (or ∠B + ∠D = 180°), then A, B, C, and D lie on a circle.
Key Takeaways for Exams:
- Memorize the definitions precisely.
- Understand the statements of all theorems and their converses.
- Be able to apply theorems to find lengths (chords, distances from centre) and angles.
- The relationship between the angle at the centre and the angle at the circumference is frequently tested.
- Properties of cyclic quadrilaterals are very important.
- Remember the special case: Angle in a semicircle is 90°.
Multiple Choice Questions (MCQs)
-
The longest chord of a circle is called its:
(A) Radius
(B) Diameter
(C) Sector
(D) Arc -
If the distance of a chord of length 16 cm from the centre of a circle of radius 10 cm is 'd', what is the value of 'd'?
(A) 6 cm
(B) 8 cm
(C) 10 cm
(D) 12 cm -
In a circle with centre O, chords AB and CD are equal. If ∠AOB = 70°, then ∠COD is:
(A) 35°
(B) 70°
(C) 110°
(D) 140° -
An arc subtends an angle of 80° at the centre. What angle will it subtend at any point on the remaining part of the circle?
(A) 80°
(B) 160°
(C) 40°
(D) 20° -
ABCD is a cyclic quadrilateral such that ∠A = 95°. What is the measure of ∠C?
(A) 85°
(B) 95°
(C) 90°
(D) 180° -
The region between a chord and its corresponding minor arc is called:
(A) Minor Sector
(B) Major Sector
(C) Minor Segment
(D) Major Segment -
If two chords of a circle are equidistant from the centre, then the chords are:
(A) Parallel
(B) Unequal
(C) Equal
(D) Perpendicular -
Angle inscribed in a semicircle is always:
(A) 60°
(B) 90°
(C) 180°
(D) 45° -
In a circle with centre O, if OM ⊥ AB and M is the midpoint of AB, which theorem is directly applicable?
(A) Theorem 10.1
(B) Theorem 10.3
(C) Theorem 10.5
(D) Theorem 10.8 -
Points P, Q, R lie on a circle. If ∠PQR = 100°, then the reflex angle ∠POR (where O is the centre) is:
(A) 100°
(B) 160°
(C) 200°
(D) 80°
Answer Key for MCQs:
- (B)
- (A) - Explanation: Perpendicular from centre bisects the chord. Half-chord = 8 cm. Use Pythagoras in the right triangle formed by radius, half-chord, and distance 'd'. 10² = 8² + d² => 100 = 64 + d² => d² = 36 => d = 6 cm.
- (B) - Explanation: Theorem 10.1 (Equal chords subtend equal angles at the centre).
- (C) - Explanation: Theorem 10.8 (Angle at circumference is half the angle at the centre).
- (A) - Explanation: Theorem 10.11 (Opposite angles of a cyclic quadrilateral sum to 180°). ∠C = 180° - 95° = 85°.
- (C)
- (C) - Explanation: Theorem 10.6 (Converse of 10.5).
- (B)
- (B) - Explanation: Theorem 10.3 states the perpendicular from the centre bisects the chord. The question describes this scenario.
- (C) - Explanation: The angle subtended by arc PR at the remaining part (Q) is 100°. By Theorem 10.8, the angle subtended by minor arc PR at the centre (∠POR) would be 2 * ∠PQR = 2 * 100° = 200°. Since ∠PQR is obtuse (100°), it must be subtended by the major arc. Therefore, the angle subtended by the minor arc PR at the centre is 2 * (180° - 100°) if Q was on the major arc, or the reflex angle is 2 * 100° = 200° if Q is on the minor arc. Given ∠PQR = 100°, this angle is subtended by the major arc PR. The angle subtended by the minor arc PR at the centre is 2 * (angle at major arc). The angle subtended by the major arc PR at the centre (reflex ∠POR) is 2 * ∠PQR = 2 * 100° = 200°.
Make sure you practice applying these theorems to various problems. Good luck with your preparation!