Chapter 9 · Class IX Mathematics · NCERT Exercises

Circles — Exercises

Chords, Angles & Cyclic Quadrilaterals — All 3 Exercises Solved

📂 3 Exercises 📝 16 Questions 🎓 High

▶ Start Exercises 📖 Revise Notes

Exercise Index

Basic terms: chord, arc, sector, segment; equal chords and their distances from centre

Perpendicular from centre bisects chord; equal chords equidistant from centre; converse theorems

Angle subtended by arc; angle at centre = 2× angle at circumference; angles in same segment equal; cyclic quadrilateral

3 exercise files · 16 total questions

Chapter at a Glance

CBSE BoardsNTSEOlympiad

11 Concepts

8 Formulas

High Difficulty

10–12% Weightage

Before You Begin

Prerequisites

✓Circle basics (Class VIII)
✓Triangle congruence (Chapter 7)
✓Quadrilateral properties (Chapter 8)

Have Ready

🔧Compass for accurate circle drawing
🔧Ruler
🔧NCERT circle theorems card

Exercise Topic Map

Exercise 9.1 Identify parts of circle; use: equal chords are equidistant from centre; perpendicular from O bisects chord

Exercise 9.2 OM ⊥ AB ⟹ AM = MB; equal chords → equal arcs; converse: equal distances from centre → equal chords

Exercise 9.3 ∠AOB = 2∠ACB (angle at centre = 2× circumference angle); angles in same segment are equal; ABCD cyclic: opp angles supplementary (∠A+∠C = 180°)

Key Formulae — Recall Before Solving

\(OM \perp AB \Rightarrow AM = MB \quad \text{(⊥ from centre bisects chord)}\)

\(\angle AOB = 2\angle ACB \quad \text{(angle at centre is double angle at circumference)}\)

\(\text{Angles in same segment are equal: } \angle ACB = \angle ADB\)

\(\angle A + \angle C = 180°;\quad \angle B + \angle D = 180° \quad \text{(Cyclic Quadrilateral)}\)

NCERT Solving Method

Step 1 — Draw the figure with centre O clearly marked; label all given points. Step 2 — Chord problems: join centre to midpoint; use the perpendicular bisector property. Step 3 — Angle theorem: identify whether the angle is at centre or on circumference; the centre angle is always 2×. Step 4 — Same segment: any two angles subtended by the same chord on the same side of it are equal. Step 5 — Cyclic quadrilateral: if question says 'cyclic', immediately note that opposite angles are supplementary.

Continue Your Preparation

📖NotesTheory & concepts 🧠MCQsTopic-wise practice ⚖️True–FalseConcept check ➡️Next ChapterHeron's Formula

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