Class 10 Mathematics Notes Chapter 9 (Chapter 9) – Examplar Problems (English) Book

Alright class, let's focus on Chapter 9: Circles from your NCERT Exemplar book. This is a crucial chapter, not just for your board exams, but also because concepts related to circles frequently appear in various government competitive exams. Pay close attention to the theorems and their applications.

Chapter 9: Circles - Detailed Notes for Exam Preparation

1. Basic Concepts (Recap):

• Circle: A collection of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).
• Radius (r): The fixed distance from the centre to any point on the circle.
• Chord: A line segment joining any two points on the circle.
• Diameter (d): The longest chord passing through the centre (d = 2r).
• Arc: A continuous piece of a circle between two points.
• Segment: The region between a chord and its corresponding arc (minor and major segments).
• Sector: The region between two radii and the corresponding arc (minor and major sectors).
• Secant: A line that intersects a circle at two distinct points.
• Tangent: A line that intersects (touches) the circle at exactly one point. This point is called the Point of Contact.

2. Tangent to a Circle:

• There is only one tangent possible at any given point on the circle.
• The tangent to a circle is a special case of the secant when the two endpoints of its corresponding chord coincide.
• Non-intersecting Line: A line that does not touch or intersect the circle.

3. Number of Tangents from a Point to a Circle:

• Point Inside the Circle: No tangents can be drawn. Any line through this point will be a secant.
• Point On the Circle: Exactly one tangent can be drawn through this point.
• Point Outside the Circle: Exactly two tangents can be drawn from this external point.

4. Key Theorems and Their Implications:

• Theorem 9.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

  • Explanation: If 'O' is the centre, 'P' is the point of contact, and XY is the tangent at P, then the radius OP is perpendicular to XY (OP ⊥ XY).
  • Significance: This theorem is fundamental for solving problems involving angles and lengths related to tangents. It often helps in forming right-angled triangles.
  • Application: Used to find angles, prove geometric properties, and solve coordinate geometry problems involving circles and tangents.

• Theorem 9.2: The lengths of tangents drawn from an external point to a circle are equal.

  • Explanation: If P is an external point and PA and PB are two tangents drawn from P to a circle with centre O, touching the circle at A and B respectively, then PA = PB.
  • Proof Outline: Join OA, OB, and OP. Consider triangles ΔOAP and ΔOBP. OA = OB (radii), OP = OP (common), ∠OAP = ∠OBP = 90° (Theorem 9.1). By RHS congruence, ΔOAP ≅ ΔOBP. Therefore, PA = PB (CPCT).
  • Corollaries/Further Results from the proof:
    • The tangents are equally inclined to the segment joining the centre to that point (∠OPA = ∠OPB).
    • The line joining the centre to the external point bisects the angle between the two tangents (OP bisects ∠APB).
    • The line joining the centre to the external point bisects the angle subtended by the points of contact at the centre (∠AOP = ∠BOP).
    • The angle subtended by the line segment joining the points of contact at the centre is supplementary to the angle between the two tangents from the point (∠AOB + ∠APB = 180°). This is very important!
  • Application: Used extensively in problems involving finding lengths, perimeters of figures involving tangents (like triangles or quadrilaterals circumscribing a circle), and proving geometric relationships.

5. Important Properties & Problem-Solving Strategies:

• Quadrilateral Circumscribing a Circle: If a quadrilateral ABCD circumscribes a circle (touches the circle at points P, Q, R, S), then the sum of opposite sides are equal.
  • AB + CD = AD + BC
  • Reasoning: Uses Theorem 9.2 repeatedly (AP=AS, BP=BQ, CR=CQ, DR=DS). Add these equations strategically.
• Parallelogram Circumscribing a Circle: A parallelogram circumscribing a circle is always a Rhombus.
  • Reasoning: Start with AB+CD = AD+BC. Since it's a parallelogram, AB=CD and AD=BC. This gives 2AB = 2AD, implying AB=AD. Since adjacent sides are equal, the parallelogram is a rhombus.
• Angle in the Alternate Segment (Often useful, though sometimes considered beyond core NCERT): The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. (Check if relevant for your specific exam syllabus).
• Pythagoras Theorem: Frequently used in conjunction with Theorem 9.1 (tangent ⊥ radius) to find lengths of tangents, radii, or distances from the centre.
• Angle Sum Property: Used for triangles and quadrilaterals formed by radii, tangents, and chords. Remember ∠AOB + ∠APB = 180° for tangents from an external point P.

Exam Focus Points:

• Direct application of Theorems 9.1 and 9.2.
• Problems involving finding lengths of tangents.
• Problems involving angles between tangents and radii.
• Properties of quadrilaterals/triangles circumscribing circles.
• Combining circle properties with properties of triangles, quadrilaterals, and Pythagoras theorem.

Multiple Choice Questions (MCQs):

1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is:
   (A) 7 cm
   (B) 12 cm
   (C) 15 cm
   (D) 24.5 cm

2. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, then ∠POA is equal to:
   (A) 50°
   (B) 60°
   (C) 70°
   (D) 80°

3. The number of tangents that can be drawn to a circle from a point inside it is:
   (A) 0
   (B) 1
   (C) 2
   (D) Infinite

4. In the figure, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to:
   (Assume a figure where PR is tangent at P, PQ is chord)
   (A) 100°
   (B) 80°
   (C) 90°
   (D) 75°

5. A parallelogram circumscribing a circle is always a:
   (A) Trapezium
   (B) Rectangle
   (C) Rhombus
   (D) Square

6. If TP and TQ are two tangents to a circle with centre O such that ∠POQ = 110°, then ∠PTQ is equal to:
   (A) 60°
   (B) 70°
   (C) 80°
   (D) 90°

7. The length of the tangent drawn from a point 8 cm away from the centre of a circle of radius 6 cm is:
   (A) √7 cm
   (B) 2√7 cm
   (C) 10 cm
   (D) 5 cm

8. PQ is a tangent to a circle with centre O at point P. If ΔOPQ is an isosceles triangle, then ∠OQP is equal to:
   (A) 30°
   (B) 45°
   (C) 60°
   (D) 90°

9. A circle touches the side BC of ΔABC at P and touches AB and AC produced at Q and R respectively. If AQ = 5 cm, then the perimeter of ΔABC is:
   (A) 5 cm
   (B) 10 cm
   (C) 15 cm
   (D) 20 cm

10. Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
    (A) 6 cm
    (B) 7 cm
    (C) 8 cm
    (D) 10 cm

Make sure you practice the problems from the Exemplar book thoroughly, as they often involve slightly more complex applications of these concepts than the standard NCERT textbook. Good luck with your preparation!