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I. Chapter Summary

This chapter explores the basic concepts and theorems related to circles, a fundamental geometric figure. Students learn about key terms like radius, chord, diameter, arc, segment, sector, and tangent, and understand important properties related to equal chords, perpendicular bisectors, and angles subtended by chords and arcs. The focus is on reasoning-based proofs and real-life applications of circle geometry.

II. Key Concepts Covered

Concept	Explanation
Circle	A set of all points in a plane equidistant from a fixed point (the centre).
Radius	The distance from the centre to any point on the circle.
Diameter	A chord that passes through the centre; longest chord of a circle.
Chord	A line segment whose endpoints lie on the circle.
Arc	A part of the circumference of a circle.
Segment	Region between a chord and an arc.
Sector	A region enclosed by two radii and the arc between them.
Perpendicular from centre to chord	Bisects the chord.
Equal chords	Are equidistant from the centre.
Angle subtended by chord at the centre	Twice the angle subtended at the remaining part of the circle.
Concyclic points	Points lying on the same circle.

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. A chord of a circle is a line segment whose endpoints lie:
   a) Outside the circle
   b) Inside the circle
   c) On the circle ✔️
   d) At the centre
2. The longest chord of a circle is:
   a) Radius
   b) Arc
   c) Diameter ✔️
   d) Sector
3. If two chords of a circle are equal, then their distance from the centre is:
   a) Equal ✔️
   b) Unequal
   c) Cannot be determined
   d) Greater than radius
4. The perpendicular drawn from the centre to a chord:
   a) Bisects the chord ✔️
   b) Divides it unequally
   c) Makes an angle of 60°
   d) Equals the chord

(B) Short Answer Questions (2/3 Marks)

1. Prove that equal chords of a circle are equidistant from the centre. (PYQ 2020)
2. In a circle with centre O, prove that the line joining the centre to the midpoint of a chord is perpendicular to the chord.
3. AB and CD are two equal chords of a circle with centre O. Show that they are equidistant from O.
4. In a circle, if a radius bisects a chord which is not a diameter, prove that it is perpendicular to the chord.

(C) Long Answer Questions (5 Marks)

1. Prove that the angle subtended by an arc at the centre is twice the angle subtended at any other part of the circle. (PYQ 2019)
2. Prove that the perpendicular from the centre of a circle to a chord bisects the chord, and use this to find the length of a chord if the radius is 13 cm and perpendicular distance from the centre is 5 cm.
3. Two chords AB and CD of a circle with centre O intersect at a point E inside the circle. If AB = CD and AE = 3 cm, EB = 5 cm, find CE and ED.
4. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to the corresponding segments of the other chord.

(D) HOTS (Higher Order Thinking Skills)

1. In a circle, prove that if the chords are equidistant from the centre, then they are equal in length.
2. Can two different chords of a circle be perpendicular bisectors of each other? Justify with a figure and reasoning.

IV. Key Formulas/Concepts

Term/Property	Definition/Statement
Angle at centre	∠AOB = 2 × ∠APB (when A, B, and P lie on circle)
Equal chords	Have equal distance from the centre
Chord bisected by radius	Perpendicular from centre bisects the chord
Diameter	Longest chord = 2 × radius
Arc length	(θ/360) × 2πr (when required)

V. Deleted Portions (CBSE 2025–2026)

 No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Chapter	Estimated Marks	Type of Questions Typically Asked
Circles	6–8 Marks	MCQ, Properties-Based Proofs, Application of Chords & Angles

VII. Previous Year Questions (PYQs)

Marks	Question	Year
2 marks	Prove that equal chords are equidistant from the centre.	PYQ 2020
3 marks	Prove that the line joining the centre to the midpoint of a chord is perpendicular to it.	PYQ 2019
5 marks	Prove that angle subtended at the centre is twice the angle subtended at the circle.	PYQ 2018

VIII. Real-World Application Examples

• Wheels and Gears: Geometry of spokes and rotation relies on circle properties.
• Clocks: Angles subtended by hands and arcs are calculated using circle geometry.
• Architecture: Circular arches, domes, and window design use chord and segment calculations.
• Engineering: Machine parts like pulleys, cogs, and flywheels rely on circle-based construction.

IX. Student Tips & Strategies for Success

 Time Management

• Day 1: Learn definitions and circle properties
• Day 2: Practice chord and angle-based theorems
• Day 3: Solve proof-based and HOTS questions

 Exam Preparation

• Use clear diagrams with labels
• Write step-by-step reasons for each geometrical statement
• Revise theorems using NCERT examples and Exemplar

 Stress Management

• Visualize circles using coins, bottle caps, wheels
• Create flashcards for each theorem and practice proofs

X. Career Guidance & Exploration

• For Classes 9–10:
  ➤ Useful in NTSE, SOF IMO, and school Olympiads
  ➤ Builds foundation for logical reasoning and construction geometry
• For Classes 11–12 & Careers:
  ➤ Relevant in:
  • Architecture
  • Mechanical Engineering
  • Industrial Design
  • Aerospace Engineering
  • Astronomy (planetary orbits)
• Exams Where Important:
  ➤ JEE (Math – Coordinate & Circle Geometry)
  ➤ CUET
  ➤ NDA
  ➤ UCEED/NID (Design Aptitude)

XI. Important Notes

 Always draw neat diagrams with all markings (O, radii, angles, chords).
 Learn theorems logically, not by rote.
 Clearly distinguish between theorems and their converse.
 Practice NCERT Exemplar problems for deeper understanding.