Answers
Exercise-wise Solutions
Find clear, step-by-step answers to the Class 9 Maths Chapter 5 exercises.
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📘 NCERT GANITA MANJARI 2026–27 • CLASS 9 MATHEMATICS
Class 9 Maths Chapter 5 Solutions
I’m Up and Down, and Round and Round
— Circles
Find Class 9 Maths Chapter 5 Solutions with clear, step-by-step answers based on the latest NCERT Ganita Manjari (2026–27). Learn the important circle concepts and theorems, understand their geometric reasoning, and quickly move to the exercise-wise solutions you need.
✓ NCERT Aligned
✓ Step-by-Step Answers
✓ Important Theorems
✓ Exercise-Wise Links
🎯 What are you looking for?
🧭 CHAPTER 5 SOLUTION HUB
Find What You Need in Chapter 5
Looking for a particular exercise answer, theorem or proof? Start with the section you need and continue learning at your own pace.
Need a solution quickly?
Choose an exercise below to reach the detailed
Class 9 Maths Chapter 5 Solutions.
Want to understand the concept first? Explore the important
circle theorems and their geometric reasoning.
Learn
Important Circle Theorems
Understand the important theorems of the chapter with diagrams, reasoning and proof-based explanations.
Explore Theorems →
Explore
Think & Reflect Solutions
Explore the additional thinking questions from Chapter 5 and follow their step-by-step solutions.
View Think & Reflect →
Revise
Quick Revision
Revise key circle concepts, important results and theorem-based ideas before solving questions.
Open Quick Revision →
Visual
Chapter 5 Knowledge Base
Use accurate geometric figures to understand circles, chords, angles and related constructions more clearly.
Learn Through Figures →
NCERT
NCERT Ganita Manjari 2026–27
Chapter 5 learning and solutions are organised around the latest NCERT Ganita Manjari textbook.
Start Chapter 5 →
💡 A simple way to use this page
Find your question → open the exercise solution → check the theorem if needed → understand the reasoning → continue with the next question.
Solution → Concept → Theorem → Practice
🎯 Chapter Learning Map
What You Will Learn in Chapter 5
From the basic language of circles to chords, arcs, angles and cyclic quadrilaterals, this chapter builds your understanding step by step through important results and theorems.
5.1
Definitions
Learn the basic terms and ideas used throughout the chapter, including the important parts and properties of a circle.
5.2
Symmetries of a Circle
Explore the symmetry of a circle and understand its rotational symmetry about the centre.
5.3
How Many Circles?
Investigate circles passing through given points and understand the uniqueness of the circle through three non-collinear points.
5.4
Chords and the Angles They Subtend
Discover the relationship between equal chords and the angles they subtend at the centre, together with the corresponding converse.
5.5
Midpoints and Perpendicular Bisectors of Chords
Understand how the centre of a circle is related to the midpoint of a chord and the perpendicular to that chord.
5.6
Distance of Chords from the Centre
Learn how the length of a chord is connected with its distance from the centre, including the important result about longer and shorter chords.
5.7
Angles Subtended by an Arc
Study angles subtended by an arc at the centre and at points on the circle, leading to the important result for an angle in a semicircle.
5.8
Concyclicity of Points
Learn when points lie on the same circle and study cyclic quadrilaterals, including the relationship between their opposite angles.
Your Chapter 5 Learning Journey
Circle Basics
→
Chords
→
Distances
→
Arcs & Angles
→
Cyclic Quadrilaterals
📐 Theorem Master Guide
Important Circle Theorems — Theorem 1 to 12
Chapter 5 is built around these important results. Read each theorem here for a quick understanding, then follow the link to learn its detailed proof step by step.
Theorem 1
Three non-collinear points
Unique Circle Through Three Non-Collinear Points
There is a unique circle passing through three non-collinear points.
Learn the Proof →
Theorem 2
Equal chords
Equal Chords and Equal Central Angles
Equal chords of a circle subtend equal angles at the centre of the circle.
Learn the Proof →
Theorem 3
Equal central angles
Equal Central Angles and Equal Chords
Chords of a circle that subtend equal angles at the centre are equal.
Learn the Proof →
Theorem 4
Centre and midpoint
Centre, Midpoint and Perpendicular to a Chord
The line joining the centre of a circle and the midpoint of a chord of the circle is perpendicular to the chord.
Learn the Proof →
Theorem 5
Perpendicular from centre
Perpendicular from the Centre Bisects a Chord
The perpendicular from the centre of a circle to a chord of the circle bisects the chord.
Learn the Proof →
Theorem 6
Equal chord lengths
Equal Chords and Equal Distances
Chords of a circle having the same length are all at the same distance from the centre of the circle.
Learn the Proof →
Theorem 7
Equal distances
Equal Distances and Equal Chords
Chords of a circle that are equidistant from the centre have equal length.
Learn the Proof →
Theorem 8
Longer chord
Longer Chord and Distance from the Centre
Let AB and DE be two chords of a circle with centre C. Suppose AB > DE. Then the distance from C to AB is less than the distance from C to DE.
Learn the Proof →
Theorem 9
Angles subtended by an arc
Angle Subtended by an Arc
The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc.
Learn the Proof →
Theorem 10
Concyclic points
Equal Angles and Concyclic Points
If a line segment AB joining two points A, B subtends equal angles at two other points C, D that lie on the same side of AB, then the four points lie on a circle.
Learn the Proof →
Theorem 11
Cyclic quadrilateral
Opposite Angles of a Cyclic Quadrilateral
The sum of two opposite angles of a cyclic quadrilateral is 180°.
Learn the Proof →
Theorem 12
Converse of Theorem 11
Converse: Opposite Angles and Concyclicity
If two opposite angles of a quadrilateral add up to 180°, then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic.
Learn the Proof →
Study Tip:
Use this section as your quick theorem guide.
For the complete proof, diagram and step-by-step reasoning,
open the relevant proof link.
💡 Think & Reflect
Think & Reflect Solutions
Strengthen your understanding of Chapter 5 through thoughtful questions that encourage reasoning, observation and deeper mathematical thinking.
Think & Reflect
Explore the Think & Reflect questions from I’m Up and Down, and Round and Round and develop a deeper understanding of circles, chords, angles and their properties.
💡 Explore Think & Reflect Solutions →
More Think & Reflect solutions are being added to this section.
Check back here as you continue your Chapter 5 preparation.
🧠 CHAPTER 5 KNOWLEDGE BASE
Chapter 5 Circles — Concept Guide
Class 9 Maths Chapter 5 में कौन-सा concept कहाँ मिलेगा? नीचे topic चुनें और सीधे उसके detailed Learn Before You Solve section पर जाएँ।
💡 How to use this guide:
यहाँ concepts को दोहराया नहीं गया है।
जिस concept को detail में सीखना हो, उसके
Learn the Concept → link पर जाएँ।
🔵
Circle Basics
Centre, radius, chord और diameter जैसे basic circle concepts से शुरुआत करें।
Learn the Concept →📍
Three Points & Circumcircle
Three non-collinear points से unique circle और circumcircle की basic idea समझें।
Learn Theorem 1 →📐
Central Angles & Chords
Equal chords और उनके द्वारा centre पर subtend किए गए angles का relationship समझें।
Learn the Concept →📏
Chord, Midpoint & Centre
Chord के midpoint, centre से perpendicular और chord की distance के बीच connection समझें।
Learn the Concept →⚖️
Equal Chords & Distance
Equal chords की centre से दूरी और उनके reciprocal relationship को समझें।
Learn the Concept →📊
Unequal Chords & Centre Distance
Longer और shorter chords की centre से दूरी का relationship तथा Theorem 8 समझें।
Learn Theorem 8 →🌙
Arcs & Angles
Minor arc, major arc और arc द्वारा centre तथा circle पर subtend किए गए angles को समझें।
Learn the Concept →🔄
Concyclic Points & Quadrilaterals
Points के concyclic होने और cyclic quadrilateral से जुड़े important ideas को revise करें।
Learn the Concept →📚
Chapter 5 Quick Revision
पूरे chapter के important ideas और theorems को quickly revise करने के लिए।
Open Quick Revision →
🎯 YOUR CHAPTER 5 LEARNING PATH
Circle Basics
→
Chords
→
Centre & Distance
→
Arcs & Angles
→
Concyclicity
📐 THEOREM REVISION MAP
Chapter 5 Theorems — 1 to 12
Circle, chords, arcs, angles और cyclic quadrilaterals से जुड़े important theorems को एक जगह revise करें। Detailed proof पढ़ने के लिए संबंधित exercise solution पर जाएँ।
💡 Revision Tip:
पहले theorem का statement समझें, फिर उसका
proof देखें। Exam में theorem लागू करते समय
उसकी conditions को ध्यान से पहचानें।
1
Circle Foundation
Theorem 1
A unique circle can be drawn through any three
non-collinear points.
Read the Proof →
2
Chords & Their Relationship with the Centre
Theorem 2
Equal chords of a circle subtend equal angles at the centre.
Proof →
Theorem 3
Chords that subtend equal angles at the centre are equal.
Proof →
Theorem 4
The line joining the centre to the midpoint of a chord is perpendicular to the chord.
Proof →
Theorem 5
The perpendicular from the centre of a circle to a chord bisects the chord.
Proof →
Theorem 6
Equal chords of a circle are equidistant from the centre.
Proof →
Theorem 7
Chords equidistant from the centre of a circle are equal.
Proof →
Theorem 8
Of two chords of a circle, the longer chord is nearer to the centre.
Proof / Explanation →
3
Arcs & Angles
Theorem 9
The angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the circle outside the arc.
Read Theorem 9 →
⭐ Important Corollary — Angle in a Semicircle
The angle subtended by a diameter at any point on the circle is 90°.
Learn This Result →
4
Concyclicity & Cyclic Quadrilaterals
Theorem 10
Equal angles subtended by AB at C and D on the same side imply that A, B, C and D are concyclic.
Proof →
Theorem 11
The sum of two opposite angles of a cyclic quadrilateral is 180°.
Proof →
Theorem 12 — Converse
If two opposite angles of a quadrilateral add to 180°, its vertices are concyclic.
Proof →
🧠 ONE-LINE REVISION
Circle
→
Chords
→
Centre & Distance
→
Arcs & Angles
→
Concyclicity
⚡ QUICK REVISION DASHBOARD
Class 9 Maths Chapter 5 — Quick Revision
Circles, Chords, Arcs, Angles & Cyclic Quadrilaterals के important concepts को जल्दी revise करें। किसी concept को detail में पढ़ना हो तो उसके Learn Before You Solve section पर जाएँ।
🎯 Before solving:
पहले इस dashboard से rule याद करें।
अगर concept में doubt हो, तो संबंधित
Learn Before You Solve link खोलें।
01 · CIRCLE BASICS
🔵 Circle, Radius, Chord & Diameter
Centre से circle के हर point की दूरी radius होती है। Centre से गुजरने वाली chord diameter होती है।
Revise from Ex 5.1 LBWS →
02 · CIRCUMCIRCLE
📍 Three Non-Collinear Points
Three non-collinear points से one unique circle गुजरता है। यही circumcircle की foundation है।
Revise Theorem 1 →
03 · CHORD RULE
⚖️ Equal Chords ↔ Equal Central Angles
Equal chords centre पर equal angles subtend करते हैं, और converse भी true है।
Revise Theorems 2 & 3 →
04 · CHORD & MIDPOINT
📏 Centre & Midpoint of a Chord
Centre को chord के midpoint से join करने वाली line chord पर perpendicular होती है।
Revise Theorems 4 & 5 →
05 · CENTRE DISTANCE
⚖️ Equal Chords ↔ Equal Distance
Equal chords centre से equal distance पर होते हैं, और converse भी true है।
Revise Theorems 6 & 7 →
06 · UNEQUAL CHORDS
📊 Longer Chord → Nearer Centre
दो chords में जो chord longer है, वह centre के closer होती है।
Revise Theorem 8 →
07 · ARCS & ANGLES
🌙 Centre Angle = 2 × Circle Angle
एक ही arc द्वारा centre पर subtended angle, circle के point पर subtended angle का double होता है।
Revise Theorem 9 →
08 · IMPORTANT RESULT
⭐ Angle in a Semicircle
Diameter द्वारा circle पर subtended angle हमेशा 90° होता है।
Revise the Corollary →
09 · CONCYCLICITY
🔄 When Are Points Concyclic?
Equal angles subtended by the same segment concyclicity establish करने में मदद करते हैं।
Revise Theorem 10 →
10 · CYCLIC QUADRILATERAL
🔷 Opposite Angles = 180°
Cyclic quadrilateral में opposite angles का sum 180° होता है।
Revise Theorem 11 →
11 · CONVERSE
🔁 Supplementary Opposite Angles
यदि quadrilateral के opposite angles का sum 180° है, तो उसके vertices concyclic होते हैं।
Revise Theorem 12 →
12 · PRACTICE
✏️ Ready to Solve?
Concepts revise हो गए? अब Chapter 5 के End-of-Chapter Exercises solve करके अपनी preparation check करें।
Start Practice →
⚡ 30-SECOND MEMORY CHECK
Equal Chords
↔
Equal Central Angles
|
Equal Chords
↔
Equal Distance
|
Longer Chord
→
Nearer Centre
|
Diameter
→
90°
|
Cyclic Quadrilateral
→
Opposite Angles = 180°
🚨 EXAM ALERT
Chapter 5 Common Exam Traps & Mistakes
Circles के questions में answer गलत होने से बचने के लिए इन common mistakes को solve करने से पहले एक बार जरूर check करें।
⚠️ Golden Rule:
सिर्फ theorem का नाम याद करना काफी नहीं है।
Question में दी गई condition देखकर सही theorem चुनें।
MISTAKE 01 · CHORDS
❌ Equal Chord ≠ Automatically Equal Distance
Equal chords की centre से equal distance बताने के लिए Theorem 6 की condition पहचानें। Theorem को बिना condition के apply न करें।
✓ Check: क्या दोनों chords equal हैं?
MISTAKE 02 · CONVERSE
🔁 Theorem 6 और 7 को उल्टा न करें
याद रखें: Equal chords → Equal distance, जबकि Equal distance → Equal chords converse direction है।
✓ Arrow की direction पहले पहचानें।
MISTAKE 03 · LONGER CHORD
📏 Longer Chord का मतलब Centre से दूर नहीं
Theorem 8 में relationship याद रखें: longer chord → nearer to the centre. Direction को उल्टा लिखना common error है।
✓ Longer = Closer
MISTAKE 04 · MIDPOINT
📐 Theorem 4 और 5 की direction न मिलाएँ
Theorem 4 में centre और midpoint of chord को join करने वाली line perpendicular होती है। Theorem 5 में centre से chord पर perpendicular chord को bisect करता है।
✓ Given क्या है और prove क्या करना है, पहले देखें।
MISTAKE 05 · THEOREM 9
🌙 Centre Angle और Circle Angle
एक ही arc के लिए centre पर angle, circle के point पर subtended angle का double होता है। दोनों angles को interchange न करें।
✓ Centre angle = 2 × angle at circle
MISTAKE 06 · SEMICIRCLE
⭐ Diameter पहचानना न भूलें
Angle in a semicircle = 90° तभी directly use करें जब given segment diameter हो।
✓ पहले देखें: क्या segment diameter है?
MISTAKE 07 · CYCLIC QUADRILATERAL
🔷 Opposite Angles को Adjacent न समझें
Cyclic quadrilateral में opposite angles का sum 180° होता है। Adjacent angles पर यह rule सीधे apply नहीं होता।
✓ Opposite + Opposite = 180°
MISTAKE 08 · CONVERSE
🔄 Theorem 11 और 12 की direction
Theorem 11 में cyclic quadrilateral से opposite angles का sum मिलता है। Theorem 12 इसका converse है: 180° sum से concyclicity establish होती है।
✓ Given → Result की direction check करें।
✅ BEFORE YOU WRITE THE FINAL ANSWER
✓ Given information सही पढ़ी?
✓ सही theorem चुना?
✓ Theorem की condition satisfy होती है?
✓ Diagram से required relation पहचाना?
✓ Calculation दोबारा check की?
✓ Final answer clearly लिखा?
YOUR CHAPTER 5 STUDY PATH
Learn → Revise → Solve → Check
Follow this simple four-step path to move from understanding the chapter to confidently checking your final preparation.
1
Learn
Build the Concepts
Start with the Chapter 5 concept guide and understand the ideas needed before solving questions.
Open Concept Guide →
2
Revise
Refresh Quickly
Use the quick revision dashboard to recall the key ideas and important relationships before practice.
Open Quick Revision →
3
Solve
Practise the Questions
Choose the required exercise and work through clear, step-by-step NCERT-aligned solutions.
Choose an Exercise →
4
Check
Test Your Preparation
Finish with the chapter-end questions to check how confidently you can apply what you have learned.
Open End Exercise →
Best approach:
Learn the concept → revise the essentials → solve the exercises →
check yourself with the chapter-end practice.
❓ FREQUENTLY ASKED QUESTIONS
Class 9 Maths Chapter 5 FAQs
Quick answers for students preparing with Class 9 Maths Chapter 5 Solutions, NCERT Ganita Manjari 2026–27, exercise-wise practice and chapter revision.
1. What is Class 9 Maths Chapter 5 about?
Class 9 Maths Chapter 5, “I’m Up and Down, and Round and Round”, introduces students to the geometry of circles. The chapter develops ideas related to circles, symmetry, chords, arcs, angles and concyclicity through definitions, activities, constructions and mathematical reasoning.
2. Where can I find Class 9 Maths Chapter 5 Solutions?
You can use the Class 9 Maths Chapter 5 Solutions on Maths Gurukulam for exercise-wise NCERT-aligned answers. The solutions are organised according to the chapter exercises so that you can directly open the exercise you are studying.
Start with the Chapter 5 Exercise Solutions section to choose the required exercise.
3. Which exercises are included in Class 9 Maths Chapter 5?
The chapter contains Exercise Sets 5.1, 5.2, 5.3, 5.4, 5.5 and 5.6, followed by the End-of-Chapter Exercises. Each exercise focuses on a different stage of understanding circles and their properties.
4. What topics are covered in Chapter 5 “I’m Up and Down, and Round and Round”?
The chapter covers definitions of a circle, symmetries of a circle, circles through given points, chords and the angles they subtend, perpendicular bisectors of chords, distance of chords from the centre, angles subtended by an arc and concyclicity of points.
5. Are the Chapter 5 solutions explained step by step?
Yes. The solutions are designed in a clear, step-by-step CBSE answer-writing style. Important reasoning, constructions and calculations are shown in a student-friendly sequence so that you can understand how an answer is obtained rather than simply copying the final result.
6. Where can I find the Think & Reflect solutions for Chapter 5?
The Think & Reflect Solutions are organised separately from the exercise-wise solutions. This makes it easier to practise the textbook’s reflective questions without mixing them with the formal exercise answers.
You can jump directly to Think & Reflect Solutions .
7. How should I prepare Class 9 Maths Chapter 5 for an exam?
A good preparation sequence is: learn the concepts → revise the key ideas → solve the exercises → attempt the chapter-end questions. Pay special attention to diagrams, construction-based questions, numerical applications and questions that require mathematical reasoning.
8. Are these Class 9 Maths Chapter 5 Solutions based on the latest NCERT Ganita Manjari?
Yes. The Chapter 5 solution material on Maths Gurukulam is prepared for the NCERT Ganita Manjari 2026–27 Class 9 Mathematics curriculum. The exercise structure and chapter terminology are aligned with the new textbook.
9. Should I solve the End-of-Chapter Exercises after completing the chapter?
Yes. The End-of-Chapter Exercises are useful for checking whether you can apply the ideas learned throughout Chapter 5. They are best attempted after completing the exercise sets and revising the chapter concepts.
10. What should I revise before solving Chapter 5 questions?
Before solving questions, revise the basic circle vocabulary, the relationship between a chord and its midpoint, the role of the centre, arcs and angles, and the conditions used in concyclicity problems. Then use the Chapter 5 Quick Revision section for a fast recap.
📚 LEARN BEYOND THE SOLUTIONS
Useful Learning Resources
Use these carefully selected resources to study Class 9 Maths Chapter 5, revise the NCERT material and explore official academic information from trusted sources.
📘
Chapter 5 Exercise Solutions
Exercise-wise answers
💡
Chapter 5 Concept Guide
Learn before you solve
⚡
Chapter 5 Quick Revision
Fast pre-exam revision
🎯
End-of-Chapter Practice
Test your preparation
📖
NCERT Official
Official academic resources ↗
🏛
CBSE Official
Official CBSE information ↗
Study Tip:
Use Maths Gurukulam for your
chapter learning, revision and solutions,
and refer to the official NCERT and CBSE websites for
official textbook and academic information.
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The Class 9 Maths Chapter 5 Solutions on Maths Gurukulam are prepared with a focus on clear mathematical reasoning, accurate presentation and student-friendly step-by-step learning. For “I’m Up and Down, and Round and Round”, the material follows the NCERT Ganita Manjari 2026–27 chapter structure and is designed to help students understand concepts, practise questions confidently and prepare effectively for school and CBSE examinations.
📘 NCERT Aligned
🎯 CBSE Focused
📝 Step-by-Step Solutions
💡 Concept-Based Learning
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✓ Student-friendly explanations
✓ NCERT-focused learning
✓ Accuracy-first solutions
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