Class 9 Maths · Original EZT practice
I'm Up and Down, and Round and Round (Class 9 Chapter 5): summary and practice
A wheel, a bangle, the sun in a cave painting - the circle is humanity's favourite shape, and this chapter takes it apart. From definitions and symmetries to chords, their distances from the centre, and the angles arcs subtend. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 22 October 2026:
1. Definitions and symmetries
A circle is all points at a fixed distance (radius) from a centre. A chord joins two points on it; the diameter, twice the radius, is the longest chord. Every diameter is a line of symmetry - infinitely many.
2. How many circles?
Through one or two points, infinitely many circles pass (centres of the latter sit on the perpendicular bisector of the pair). Through three non-collinear points, exactly one - its centre is where the perpendicular bisectors meet.
3. Chords and their perpendicular bisectors
Equal chords subtend equal angles at the centre (SSS congruence, since all radii are equal), and conversely. The perpendicular from the centre to a chord bisects it, and the line from the centre to a chord's midpoint is perpendicular to it.
4. Distance from centre; angles subtended by arcs
Equal chords are equidistant from the centre; of unequal chords the longer is closer, and the diameter has distance zero. Headline result: the angle an arc subtends at the centre is twice what it subtends at any point on the rest of the circle - even when the centre angle is reflex.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Relate diameter and radius
Choose the method: The diameter is the chord through the centre.
EZT practice question
How is the diameter related to the radius, and what is special about it among chords?
Show the worked solution
- The diameter passes through the centre, so it is two radii end to end.
- Diameter = 2 x radius.
- No chord can be longer - any chord off the centre is shorter.
Answer: Diameter = 2 x radius; it is the longest possible chord.
Check: Its distance from the centre is zero - the closest a chord can get - matching the rule that longer chords sit closer to the centre.
Watch for this: Do not say a chord can exceed the diameter - the diameter is the maximum.
Now try it yourself
Radius 6.5 cm - diameter?
Check your answer
13 cm.
2 x 6.5 = 13.
Count the symmetries
Choose the method: Every diameter folds the circle onto itself.
EZT practice question
How many lines of symmetry does a circle have?
Show the worked solution
- Fold along any diameter and the two halves match exactly.
- There are infinitely many diameters.
- So there are infinitely many lines of symmetry - and rotation about the centre by any angle also maps the circle to itself.
Answer: Infinitely many - every diameter is one.
Check: This perfect symmetry is why all radii are equal, and it powers every chord result in the chapter.
Watch for this: Do not answer 2 or 4 - those are rectangle and square numbers; a circle has no preferred direction.
Now try it yourself
Does a circle look the same after a 37-degree rotation about its centre?
Check your answer
Yes.
Any rotation about the centre maps it to itself.
Circles through given points
Choose the method: Centres equidistant from two points lie on their perpendicular bisector.
EZT practice question
How many circles pass through three points that are not in a line?
Show the worked solution
- The centre must be equidistant from all three points.
- Centres equidistant from two of them lie on that pair's perpendicular bisector.
- The two perpendicular bisectors meet in exactly one point - so exactly one circle.
Answer: Exactly one.
Check: Through one point or two points there are infinitely many circles; the third point pins the centre to a single intersection.
Watch for this: Three COLLINEAR points give no circle - the bisectors run parallel.
Now try it yourself
Where can the centre of a circle through two given points lie?
Check your answer
Anywhere on their perpendicular bisector.
Every point of that line is equidistant from the two.
Equal chords, equal angles
Choose the method: Pair the triangles made with the centre.
EZT practice question
Chords AB and DE of the same circle are equal. What can you say about the angles they subtend at the centre?
Show the worked solution
- Join the centre O to A, B, D, E - all radii, all equal.
- Triangles OAB and ODE have three equal sides: SSS congruent.
- So angle AOB equals angle DOE.
Answer: They are equal.
Check: The converse also holds: chords subtending equal angles at the centre are equal - the same congruence run backwards.
Watch for this: The chords must belong to the SAME circle (or equal circles) - equal radii are the engine of the proof.
Now try it yourself
Two chords subtend equal angles at the centre. Are the chords equal?
Check your answer
Yes.
The converse of the same result.
Perpendicular from centre to chord
Choose the method: The perpendicular from the centre bisects the chord.
EZT practice question
A chord is 16 cm long and the perpendicular from the centre meets it at M. What is AM?
Show the worked solution
- The perpendicular from the centre to a chord bisects the chord.
- So M is the midpoint.
- AM = 16 / 2 = 8 cm.
Answer: 8 cm.
Check: Why it works: the centre is equidistant from the chord's endpoints, so it lies on the chord's perpendicular bisector. The converse holds too - the line from the centre to the midpoint is perpendicular to the chord.
Watch for this: Do not halve only when told M is the midpoint - the perpendicular from the centre GUARANTEES the midpoint.
Now try it yourself
The centre is joined to a chord's midpoint. At what angle does it meet the chord?
Check your answer
90 degrees.
The converse statement.
Angle subtended by an arc
Choose the method: Centre angle = twice the circumference angle.
EZT practice question
An arc subtends 100 degrees at the centre. What angle does it subtend at a point on the rest of the circle? And if the centre angle is a reflex 260 degrees?
Show the worked solution
- The angle at the centre is twice the angle at any point on the remaining part.
- 100 / 2 = 50 degrees.
- The rule holds for reflex angles too: 260 / 2 = 130 degrees.
Answer: 50 degrees; 130 degrees.
Check: Under 180 degrees at the centre means a minor arc; over 180 degrees means major - in both cases halving gives the circumference angle.
Watch for this: Do not double - the centre angle is the BIGGER one; halve it for the circumference.
Now try it yourself
An arc subtends 80 degrees at the centre. Angle at the circumference?
Check your answer
40 degrees.
80 / 2 = 40.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Chord?
Show answer and reason
A segment joining two points on the circle.
The diameter is the longest chord, twice the radius.
2. Perpendicular from centre to chord?
Show answer and reason
It bisects the chord.
The centre lies on the chord's perpendicular bisector.
3. Which chord is closer to the centre?
Show answer and reason
The longer one.
Equal chords are equidistant; the diameter has distance zero.
4. Arc angle at centre vs circumference?
Show answer and reason
Centre angle is double.
Holds even for reflex centre angles.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Explain why exactly one circle passes through three non-collinear points.
- A chord is 24 cm long. How far is its midpoint from each endpoint?
- An arc subtends 80 degrees at the centre - find the angle at the circumference.
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Questions about this practice page
- What is a chord of a circle?
- A line segment joining two points on the circle. The diameter is the longest chord and equals twice the radius.
- What does the perpendicular from the centre do to a chord?
- It bisects the chord. Conversely, the line from the centre to a chord's midpoint is perpendicular to the chord - the centre is equidistant from the chord's endpoints.
- Which chord is closer to the centre - a longer or a shorter one?
- The longer chord. Equal chords are equidistant from the centre, and the diameter, the longest chord of all, has distance zero because it passes through the centre.
- What is the angle-subtended-by-an-arc theorem?
- The angle an arc subtends at the centre is twice the angle it subtends at any point on the remaining part of the circle - even when the centre angle is reflex.
- Are these the NCERT exercise questions for Class 9 Maths Chapter 5?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.