This MCQ module is based on: Introduction to Circles
TOPIC 13 OF 25
Introduction to Circles
🎓 Class 9
Mathematics
CBSE
Theory
Ch 5 — I’m Up and Down, and Round and Round
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Introduction to Circles
Targeting Class 9 level in Geometry, with Intermediate difficulty.
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5.1 Circles Around Us
Humans have always been captivated by the shapes that surround us. The earliest cave paintings — including those from Gorakhandi in Odisha — already show round and oval geometric patterns: triangles, squares, circles and ovals. Circles seem to have inspired humans long before language did.
Circles appear everywhere in nature. Ripples on still water move outward as perfect circles. The cross-section of a tree trunk or a sunflower's seed-bearing disc is circular. The full Moon? and the Sun (during a total solar eclipse) appear circular too.
Ripples
Disturbing water spreads waves in widening circles around the point of impact.
Sunflower
The disc that bears the seeds is almost perfectly circular.
The Moon
A full Moon appears as a bright disc in the night sky.
Architecture
Wheels, domes, arches and rangoli use circular forms for strength and beauty.
5.2 Symmetries of a Circle
Take a paper plate (or trace a bangle). Cut along any straight line that passes through the centre. Both halves overlap exactly — so the line is a line of symmetry. In fact, every straight line through the centre is a line of symmetry, and a circle has infinitely many lines of reflection symmetry.
Fig. 5.4 — Every line through the centre is a line of symmetry.
Definition — Diameter
The line segment joining two points on a circle that passes through the centre is called a diameter. All diameters of a circle are lines of symmetry.Think and Reflect
1. What is the rotational symmetry of a square? How many lines of reflection symmetry does it have? What about a regular hexagon?2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
5.3 How Many Circles?
How many circles can be drawn through one given point? Through two points? Through three?
- One point A: Infinitely many circles pass through A — every centre at every distance gives a different circle.
- Two points A and B: Infinitely many. The centre of any such circle is equidistant from A and B, so it lies on the perpendicular bisector of AB.
- Three points A, B and C: If the three points are collinear, no circle can pass through all three. If they are non-collinear, exactly one circle passes through them.
Theorem 1
There is one and only one circle passing through three given non-collinear points.Why? Let O be the centre of such a circle. Then OA = OB, so O lies on the perpendicular bisector of AB. Similarly, O lies on the perpendicular bisector of BC. The two perpendicular bisectors meet at exactly one point because the points are non-collinear. So the centre — and hence the circle — is unique.
Fig. 5.5 — The circumcircle of triangle ABC.
5.4 Parts of a Circle
Once a circle is drawn we can identify many useful parts. Each has a precise name and definition.
Fig. 5.6 — Centre O, radius OP, diameter QP, chord AB.
Key Definitions
Circle: the set of all points in a plane at a fixed distance from a fixed point, called the centre.Radius: the fixed distance from the centre to any point on the circle (also the segment from centre to the circle).
Chord: a line segment whose endpoints are on the circle.
Diameter: a chord that passes through the centre. Diameter \(= 2 \times\) radius.
Arc: a continuous portion of the circle. Two points on a circle divide it into a minor arc and a major arc.
Sector: the region bounded by two radii and the arc between them (like a pizza slice).
Segment: the region bounded by a chord and an arc.
A sector (left) is bounded by two radii and an arc. A segment (right) is bounded by a chord and an arc.
Try this: The interior of a circle is called a disc. Is a diameter a chord? Yes — and it is the longest possible chord of the circle.
Concentric, Congruent and Equal Circles
Concentric circles share the same centre but have different radii — like the rings on a target board. Congruent circles have equal radii. They can be slid (translated) to overlap exactly. Two circles are equal precisely when they have the same radius.
Activity: Folding to Find the Centre
L3 ApplyMaterials: a circular paper cut-out (any size), pencil.
Predict: If you fold the disc in half, then in half again, where do the creases meet?
- Fold the disc so that the curved edge meets itself — a sharp half-fold.
- Open it. The crease is a diameter.
- Fold again along a different direction. Open. A second diameter appears.
- Mark the point where the two creases cross.
The crossing point is the centre of the disc. Every diameter passes through the centre, so two non-parallel diameters always intersect there. This is one of the easiest ways to find a circle's centre experimentally.
5.5 Worked Examples
Example 1. The diameter of a circular pond is 14 m. What is its radius? If a duck swims along the boundary once, how far has it travelled? (Take \(\pi=\tfrac{22}{7}\).)
Radius \(r = \tfrac{14}{2} = 7\) m. Distance once around \(=\) circumference \(= 2\pi r = 2 \times \tfrac{22}{7} \times 7 = 44\) m.
Example 2. Three points A, B, C lie on a sheet. AB = 5 cm, BC = 6 cm and AC = 7 cm. Can a circle be drawn through all three? If yes, why exactly one?
Since 5 + 6 > 7, the three points are non-collinear (they form a genuine triangle). By Theorem 1, exactly one circle — the circumcircle — passes through them. Its centre is the intersection of the perpendicular bisectors of any two of AB, BC and AC.
Historical Note
Ancient civilisations associated the circle with perfection. The Sanskrit word for circle, vṛtta, also means "completed" or "whole". Aryabhata (5th century CE) used the circle to define precise values of \(\sin\) and \(\cos\), and gave one of the earliest accurate approximations of \(\pi\) (\(\pi \approx 3.1416\)).Competency-Based Questions
Scenario: A community park is being designed around a circular fountain at the centre. The municipal architect wants to place three benches A, B and C so that each bench is the same distance from the fountain. The current sketch shows benches A and B already placed; bench C still needs a position.
Q1. The architect wants every position on a curved walking track to be the same distance from the fountain. What shape must the track be?
L3 ApplyAnswer: (c). The set of all points at a fixed distance from a fixed point (the fountain) is, by definition, a circle.
Q2. Bench C must be on the same circular track as A and B. Analyse: how many positions for C are possible if the track has not yet been drawn?
L4 AnalyseAnswer: Through any two points A and B infinitely many circles pass — their centres lie on the perpendicular bisector of AB. Every such circle gives valid positions for C. So infinitely many positions are possible until a third constraint (the radius, or the centre) is fixed.
Q3. The architect now fixes the fountain at point O. Evaluate the claim: "Once O is fixed, there are still many circles through A and B with centre O."
L5 EvaluateAnswer: The claim is false. With centre O fixed, the radius must equal OA and OB. So OA must equal OB — i.e. A and B must already be equidistant from O — and then exactly one circle exists. If OA ≠ OB no such circle exists.
Q4. Design: place benches so they form a triangle around the fountain such that the fountain is the circle's centre AND the triangle is equilateral with side 6 m. What is the radius of the bench-circle? (\(\sqrt{3}\approx1.732\))
L6 CreateAnswer: For an equilateral triangle of side \(a\), the circumradius is \(R = \tfrac{a}{\sqrt 3}\). With \(a=6\): \(R = \tfrac{6}{\sqrt 3} = 2\sqrt 3 \approx 3.46\) m. Place the three benches at 120° apart on this circle.
Assertion–Reason Questions
Assertion (A): A diameter is the longest chord of a circle.
Reason (R): A diameter passes through the centre.
Reason (R): A diameter passes through the centre.
Answer: (a). Any chord can be at most twice the radius (the distance between two points in a disc); equality occurs precisely when the chord passes through the centre, i.e. when it is a diameter. So R explains A.
Assertion (A): One and only one circle passes through three given collinear points.
Reason (R): The perpendicular bisectors of two segments always meet at a unique point.
Reason (R): The perpendicular bisectors of two segments always meet at a unique point.
Answer: (d). A is false: through three collinear points NO circle exists. R is true in general but applies only to non-collinear points; with collinear points the bisectors are parallel.
Assertion (A): A circle has infinitely many lines of reflection symmetry.
Reason (R): Every diameter is a line of reflection symmetry of the circle.
Reason (R): Every diameter is a line of reflection symmetry of the circle.
Answer: (a). A circle has infinitely many diameters, and each is a line of symmetry. So R directly explains A.
Frequently Asked Questions
What is a circle in Class 9 Maths Chapter 5?
A circle is the set of all points in a plane that are at a fixed distance, called the radius, from a fixed point called the centre. The boundary itself is the circle, while the region it encloses is the circular disc.
What is the difference between a chord and a diameter?
A chord is a line segment whose two endpoints lie on the circle. A diameter is a special chord that passes through the centre. Every diameter is a chord, but only the longest chord — and any chord through the centre — is a diameter.
What is the difference between an arc and a chord?
A chord is the straight segment between two points on the circle. An arc is the curved path along the circle between the same two points. Every chord (other than a diameter) divides the circle into two unequal arcs: a major arc and a minor arc.
What is the difference between a sector and a segment of a circle?
A sector is the pizza-slice region bounded by two radii and the arc between them. A segment is the region bounded by a chord and the arc on one side of the chord. A sector includes the centre; a segment usually does not.
What is the relationship between radius and diameter of a circle?
The diameter d is exactly twice the radius r, so d = 2r and r = d/2. Both are measured in the same units, and both pass through the centre when drawn from boundary to boundary.
How are circles used in real life?
Wheels, gears, clock faces, pizza slices, satellite orbits, athletic tracks and the cross-section of pipes all use circles because their geometric properties — equal radii, smooth boundary, and symmetry — make rolling, sealing and uniform distribution natural.
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Mathematics Class 9 — Ganita Manjari
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