Chapter 5 of 6

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Rajasthan (RBSE)Class 9 Mathematics★ 5–7 marks · RBSE Class 9 Mathematics (new NCF — Ganita Prakash 9)

Circles — Symmetry, Chords and Constructions

The new NCF Class 9 Maths chapter (titled "I'm Up and Down, and Round and Round"). Circle vocabulary — centre, radius, chord, diameter, arc, sector, segment; the symmetries of a circle; chord properties (the perpendicular from the centre bisects a chord; equal chords are equidistant); and how many circles pass through given points — with solved board-pattern problems.

⏱ 30 min readMedium difficulty✓ Free · NCERT-aligned

Worksheet

Podcast

Listen as a conversation

A teacher & a student talk through the chapter — in your language.

Mind Map

The chapter at a glance

Every concept and exam-favourite branch — revisable in 5 minutes.

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Study Guide

Night-before notes

Key points, glossary and the questions most likely to show up.

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By the end of this chapter you'll be able to…

1Define the circle and its parts: radius, chord, diameter, arc, secant, tangent, sector, segment
2Describe the line and rotational symmetry of a circle
3State and use the chord properties (perpendicular from centre bisects a chord; equal chords equidistant)
4Apply r² = d² + ℓ² to relate radius, centre-distance and chord length
5Explain why exactly one circle passes through three non-collinear points and construct it

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Why this chapter matters

Circle properties are pure-geometry favourites in the RBSE paper — chord theorems and the 'distance–radius–half-chord' Pythagoras link give reliable 2–3 mark questions, and the constructions (circle through three points) test reasoning. It is the foundation for the Class 10 Circles (tangents) chapter.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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The Use of Coordinates (Class 9)

Perpendicular bisectors and locating points.

Circles — Symmetry, Chords and Constructions (RBSE Class 9 · Mathematics)

The circle is the most symmetric shape there is — turn it any amount about its centre and it looks unchanged. The new book captures that with a playful title, "I'm Up and Down, and Round and Round." This chapter turns that symmetry into precise, provable facts about chords and constructions.

RBSE note (2026-27). Class 9 uses the new NCF (Ganita Prakash 9) Mathematics textbook; this chapter's book title is "I'm Up and Down, and Round and Round." BSER (Ajmer) sets the exam.

1. Circle vocabulary

A circle is the set of all points in a plane at a fixed distance (the radius) from a fixed point (the centre).

Term	Meaning
Radius (r)	centre to any point on the circle
Chord	a segment joining two points on the circle
Diameter (d)	the longest chord; passes through the centre; d = 2r
Arc	a part of the circle (minor / major)
Secant	a line cutting the circle at two points
Tangent	a line touching the circle at one point
Sector	region between two radii and an arc
Segment	region between a chord and an arc

The interior + the circle + the exterior partition the plane.

2. Symmetries of a circle

Line (reflective) symmetry: every line through the centre is a line of symmetry — a circle has infinitely many.
Rotational symmetry: a circle maps onto itself under any angle of rotation about its centre (infinite-order rotational symmetry).

This symmetry is the reason behind every chord theorem below.

3. Chord properties (provable from symmetry)

The perpendicular from the centre to a chord bisects the chord. (And conversely, the line from the centre to the midpoint of a chord is perpendicular to it.)
The perpendicular bisector of a chord passes through the centre. — This is how you find an unknown centre.
Equal chords are equidistant from the centre (and chords equidistant from the centre are equal).

If a chord of length $2 ℓ$ is at distance $d$ from the centre of a circle of radius $r$ , then by Pythagoras: $r^{2} = d^{2} + ℓ^{2} .$

4. How many circles pass through given points?

Through one point: infinitely many circles.
Through two points A and B: infinitely many — all their centres lie on the perpendicular bisector of AB.
Through three non-collinear points: exactly one circle. Its centre is where the perpendicular bisectors of two of the chords meet (this is the circumcentre).
Through three collinear points: no circle.

This gives a construction: to draw the unique circle through three points, draw the perpendicular bisectors of two pairs; their intersection is the centre, and its distance to any of the points is the radius.

5. Worked example

A chord of a circle of radius 5 cm is at a distance of 3 cm from the centre. Find the length of the chord.

Half-chord $ℓ$ : $r^{2} = d^{2} + ℓ^{2} \Rightarrow 5^{2} = 3^{2} + ℓ^{2} \Rightarrow ℓ^{2} = 25 - 9 = 16 \Rightarrow ℓ = 4$ .

Chord length $= 2 ℓ = 2 \times 4 =$ 8 cm.

6. Quick recap

A circle = points at fixed distance r from the centre; diameter d = 2r is the longest chord.
Know chord, arc, secant, tangent, sector, segment.
A circle has infinitely many lines of symmetry (all through the centre) and infinite rotational symmetry.
Perpendicular from centre bisects a chord; the perpendicular bisector of a chord passes through the centre; equal chords are equidistant from the centre.
Use $r^{2} = d^{2} + ℓ^{2}$ to link radius, centre-distance and half-chord.
Exactly one circle passes through three non-collinear points (centre = circumcentre).

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Diameter

d = 2r

Longest chord; passes through the centre.

Chord–distance relation

r² = d² + ℓ²

d = centre-to-chord distance, ℓ = half the chord length.

Chord length

chord = 2ℓ = 2√(r² − d²)

From the relation above.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using the full chord instead of half in r² = d² + ℓ²

✓ ℓ is HALF the chord, because the perpendicular from the centre bisects it. Chord length = 2ℓ.

WATCH OUT

✗ Thinking a chord and a diameter are different kinds of things

✓ A diameter IS a chord — the special longest one that passes through the centre.

WATCH OUT

✗ Saying a circle has 4 lines of symmetry

✓ A circle has INFINITELY many lines of symmetry — every line through the centre.

WATCH OUT

✗ Believing a circle can pass through any three points

✓ Only three NON-collinear points give a (unique) circle. Three collinear points give none.

WATCH OUT

✗ Forgetting the perpendicular bisector of a chord passes through the centre

✓ This is the key fact used to locate an unknown centre — draw two and intersect them.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

In-text (definitions & symmetry)

Parts of a circle; lines of symmetry; longest chord

Questions

In-text (chords)

Perpendicular from centre; equal chords; r² = d² + ℓ²

Questions

In-text (how many circles)

Circles through 1, 2, 3 points; circumcentre

Questions

End-of-chapter Ex

End-of-chapter exercise

Chord numericals, proofs, constructions

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Vocabulary

What is the longest chord of a circle called?

▶ Show solution

The diameter (it passes through the centre and equals 2r).

✦ Answer: the diameter.

Q2EASY· Symmetry

How many lines of symmetry does a circle have?

▶ Show solution

Infinitely many — every line through the centre.

✦ Answer: infinitely many.

Q3EASY· Diameter

A circle has radius 7 cm. What is its diameter?

▶ Show solution

d = 2r = 2 × 7 = 14 cm.

✦ Answer: 14 cm.

Q4MEDIUM· Chord

A chord of length 24 cm is drawn in a circle of radius 13 cm. How far is it from the centre?

▶ Show solution

Step 1 — half-chord ℓ = 12 cm.

Step 2 — r² = d² + ℓ² → 13² = d² + 12² → 169 = d² + 144.

Step 3 — d² = 25 → d = 5 cm.

✦ Answer: 5 cm.

Q5MEDIUM· Chord

A chord is 8 cm from the centre of a circle of radius 10 cm. Find the chord's length.

▶ Show solution

Step 1 — ℓ² = r² − d² = 100 − 64 = 36 → ℓ = 6.

Step 2 — chord = 2ℓ = 12 cm.

✦ Answer: 12 cm.

Q6MEDIUM· Property

State the property used to locate the centre of a circle from a chord.

▶ Show solution

The perpendicular bisector of any chord passes through the centre. Drawing the perpendicular bisectors of two chords; their intersection is the centre.

✦ Answer: perpendicular bisector of a chord passes through the centre.

Q7HARD· How many circles

How many circles pass through three given points? Explain for non-collinear and collinear cases.

▶ Show solution

Step 1 — Three NON-collinear points: exactly ONE circle; its centre is the intersection of the perpendicular bisectors of two of the chords (the circumcentre).

Step 2 — Three COLLINEAR points: the perpendicular bisectors are parallel and never meet, so NO circle passes through them.

✦ Answer: one (non-collinear); none (collinear).

Q8HARD· Equal chords

Two equal chords of a circle of radius 17 cm are each 15 cm long. Show they are equidistant from the centre and find that distance.

▶ Show solution

Step 1 — Equal chords are equidistant from the centre (chord theorem), so both have the same d.

Step 2 — ℓ = 15/2 = 7.5 cm; d² = r² − ℓ² = 289 − 56.25 = 232.75.

Step 3 — d = √232.75 ≈ 15.26 cm.

✦ Answer: equidistant; d ≈ 15.26 cm.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Circle = points at distance r from the centre; diameter d = 2r is the longest chord.
•Parts: chord, arc, secant, tangent, sector, segment.
•Symmetry: infinitely many lines of symmetry (through the centre) and infinite rotational symmetry.
•Perpendicular from the centre bisects a chord; perpendicular bisector of a chord passes through the centre.
•Equal chords are equidistant from the centre (and vice versa).
•r² = d² + ℓ² (ℓ = half-chord); chord = 2√(r² − d²).
•Exactly one circle through three non-collinear points (circumcentre); none if collinear.

Rajasthan (RBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 5–7 marks

Question type	Marks each	Typical count	What it tests
MCQ / fill in the blank	1	1–2	Vocabulary, symmetry, diameter
Short answer / numerical	2	2	Chord–distance with r² = d² + ℓ²; properties
Short answer + proof/construction	3	1	How-many-circles reasoning; equal chords; construction

Prep strategy

Memorise the chord theorems and the r² = d² + ℓ² relation
Always halve the chord before using Pythagoras
Practise the 'circle through three points' construction and reasoning
Draw a neat figure for every chord/centre question

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Finding a wheel's centre

Workshops use the perpendicular-bisector-of-a-chord trick to locate the exact centre of a disc.

Architecture & arches

Circular arcs and their chords are designed using these exact relations.

GPS trilateration

Locating a point as the intersection of circles of known radius is the circumcircle idea in action.

Design & manufacturing

Symmetry of the circle underlies gears, lenses and turntables.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Draw the figure and drop the perpendicular from the centre to the chord first.
Use r² = d² + ℓ² with ℓ = half the chord; never the full chord.
For 'how many circles', argue via perpendicular bisectors of chords.
Quote the chord theorem you use as the first line of a proof.
State units and keep the figure labelled for method marks.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Angle subtended by a chord at the centre vs on the circle (Class 10 preview).
Cyclic quadrilaterals and the circumcircle of a triangle.
Power of a point and intersecting-chords theorem.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

RBSE Class 9 Annual (BSER Ajmer)High — chord numericals + a proof each year

NTSE / NMMSMedium — circle-property MCQs

JEE FoundationHigh — base for Class 10 Circles (tangents)

Maths Olympiad (IMO)Medium-high — circle geometry is a staple

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

It is a playful nod to a circle's perfect symmetry — it looks the same however you flip it (up/down) or turn it (round and round). The maths content is standard circle geometry.

Yes — a diameter is the longest chord, the one that passes through the centre. Every diameter is a chord, but not every chord is a diameter.

Because the perpendicular from the centre bisects the chord, the right triangle has legs d (centre to chord) and ℓ (half the chord), with the radius r as hypotenuse.

Draw any two chords, construct their perpendicular bisectors, and where the bisectors meet is the centre (both bisectors pass through it).

Practice more, ask doubts, find a tutor

Rajasthan (RBSE) Class 9 mock tests

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