Two chords AB and CD of a circle are equal in length. Which of the following is true?

They subtend equal angles at the centre

They subtend different angles at the centre

They subtend equal angles at the centre

One chord is longer than the other

Assertion (A): In a circle, equal chords subtend equal angles at the centre. Reason (R): If two chords subtend equal angles at the centre of a circle, then the chords are equal.

Both A and R are true but R is not the correct explanation of A.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

In a circle of radius 25 cm, a chord is drawn at a distance of 24 cm from the centre. Find the length of the chord. If another chord in the same circle is 30 cm long, find its distance from the centre.

Length of chord = 14 cm; distance of 30 cm chord from centre = 20 cm.

Given two points A and B that are 8 cm apart, describe the set of all possible centres of circles that pass through both A and B. If a circle passing through A and B has a radius of 5 cm, calculate the distance of its centre from the line segment AB. Also, determine the smallest possible radius of a circle through A and B.

The centres lie on the perpendicular bisector of AB. For radius 5 cm, the distance from centre to AB is 3 cm. The smallest radius is 4 cm.

Prove that there is exactly one circle passing through three non-collinear points. Hence, show how the circumcentre of triangle ABC is located using perpendicular bisectors. What happens to the circumcentre when the triangle is right-angled?

There is exactly one circle (the circumcircle) through non-collinear points A, B, C. The circumcentre O is the intersection of the perpendicular bisectors of any two sides. For a right-angled triangle, the circumcentre is the midpoint of the hypotenuse.

A circular park has two straight walking paths represented by chords AB and CD. Both paths have the same length. The centre of the park is marked as O, and lines OA, OB, OC, and OD are drawn. (1) What is the relationship between ∠AOB and ∠COD? Explain. (2) If ∠AOB = 54°, find ∠COD. (3) If another chord EF of the same length as AB is drawn, what angle will it subtend at the centre? Give a reason.

(i) They are equal; equal chords subtend equal angles. (ii) 54°. (iii) 54°; by the same theorem.

Class 9 Maths Chapter 5: I'm Up and Down, and Round and Round — Important Questions & Sample Paper

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Yes — this page has 44+ original Class 9 Mathematics Chapter 5 (“I'm Up and Down, and Round and Round”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 5 'I'm Up and Down, and Round and Round' from the NCERT Ganita Manjari textbook for Class 9 explores the fascinating world of circles. Starting with the basic definition—a set of points equidistant from a fixed center—students learn about key elements like radius, diameter, chord, arc, and segment. The chapter then dives into fundamental properties: the perpendicular from the center to a chord bisects the chord, and its converse; equal chords are equidistant from the center, and they subtend equal angles at the center. A circle's perfect symmetry is highlighted, with infinitely many lines of reflection through the center. Problem-solving often involves applying the Pythagorean theorem to find chord lengths or distances from the center. For instance, given the radius and the distance of a chord from the center, the chord length can be calculated. Another key concept is the circumcircle of a triangle—the circle passing through all three vertices. For a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse, making the hypotenuse the diameter. Exam questions frequently test these ideas: calculating missing chord lengths, identifying symmetries, constructing circles through points, and reasoning about equal chords. By mastering this chapter, students build a strong foundation for more advanced geometry in higher classes.

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Mathematics — I'm Up and Down, and Round and Round

Class 9Time: 3 hrsMax Marks: 80

SECTION A

1.
Two chords AB and CD of a circle are equal in length. Which of the following is true?
(a) They subtend different angles at the centre(b) They subtend equal angles at the centre(c) One chord is longer than the other(d) They are parallel
1
2.
In a circle, a chord of length 16 cm is at a distance of 6 cm from the centre. Another chord is at a distance of 8 cm from the centre. What is the length of the other chord?
(a) 12 cm(b) 14 cm(c) 16 cm(d) 20 cm
1
3.
A chord of a circle that passes through its centre is called the:
(a) Radius(b) Diameter(c) Tangent(d) Secant
1

+ 41 more questions in the full paper

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

Real, exam-style questions to practise and revise — each with its answer. Generate a full paper for unlimited more.

Q1. Two chords AB and CD of a circle are equal in length. Which of the following is true?
1 mark
Multiple Choice (MCQ)
(A) They subtend different angles at the centre(B) They subtend equal angles at the centre(C) One chord is longer than the other(D) They are parallel
▸ Answer▾ Answer
They subtend equal angles at the centre
Q2. In a circle, a chord of length 16 cm is at a distance of 6 cm from the centre. Another chord is at a distance of 8 cm from the centre. What is the length of the other chord?
1 mark
Multiple Choice (MCQ)
(A) 12 cm(B) 14 cm(C) 16 cm(D) 20 cm
▸ Answer▾ Answer
12 cm
Q3. A chord of a circle that passes through its centre is called the:
1 mark
Multiple Choice (MCQ)
(A) Radius(B) Diameter(C) Tangent(D) Secant
▸ Answer▾ Answer
Diameter
Q4. If two chords of a circle subtend equal angles at the centre, then which of the following is true?
1 mark
Multiple Choice (MCQ)
(A) The chords are unequal(B) The chords are equal(C) One chord is a diameter(D) The chords are perpendicular
▸ Answer▾ Answer
The chords are equal
Q5. Assertion (A): In a circle, equal chords subtend equal angles at the centre. Reason (R): If two chords subtend equal angles at the centre of a circle, then the chords are equal.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
Both A and R are true but R is not the correct explanation of A.
Q6. In a circle, chords PQ and RS each subtend an angle of 75° at the centre. Are PQ and RS equal? Justify.
2 marks
Short Answer
▸ Answer▾ Answer
Yes, PQ and RS are equal.
Q7. The longest chord of a circle is 20 cm. What is the radius of the circle?
2 marks
Short Answer
▸ Answer▾ Answer
10 cm
Q8. In a circle of radius 25 cm, a chord is drawn at a distance of 24 cm from the centre. Find the length of the chord. If another chord in the same circle is 30 cm long, find its distance from the centre.
3 marks
Short Answer
▸ Answer▾ Answer
Length of chord = 14 cm; distance of 30 cm chord from centre = 20 cm.
Q9. A chord of a circle is 18 cm long and is at a distance of 12 cm from the centre. Find the radius of the circle.
3 marks
Short Answer
▸ Answer▾ Answer
15 cm
Q10. Given two points A and B that are 8 cm apart, describe the set of all possible centres of circles that pass through both A and B. If a circle passing through A and B has a radius of 5 cm, calculate the distance of its centre from the line segment AB. Also, determine the smallest possible radius of a circle through A and B.
5 marks
Long Answer
▸ Answer▾ Answer
The centres lie on the perpendicular bisector of AB. For radius 5 cm, the distance from centre to AB is 3 cm. The smallest radius is 4 cm.
Q11. Prove that there is exactly one circle passing through three non-collinear points. Hence, show how the circumcentre of triangle ABC is located using perpendicular bisectors. What happens to the circumcentre when the triangle is right-angled?
5 marks
Long Answer
▸ Answer▾ Answer
There is exactly one circle (the circumcircle) through non-collinear points A, B, C. The circumcentre O is the intersection of the perpendicular bisectors of any two sides. For a right-angled triangle, the circumcentre is the midpoint of the hypotenuse.
Q12. A circular park has two straight walking paths represented by chords AB and CD. Both paths have the same length. The centre of the park is marked as O, and lines OA, OB, OC, and OD are drawn.
4 marks
Case Study
1. (i) What is the relationship between ∠AOB and ∠COD? Explain.2 marks
2. (ii) If ∠AOB = 54°, find ∠COD.1 mark
3. (iii) If another chord EF of the same length as AB is drawn, what angle will it subtend at the centre? Give a reason.1 mark
▸ Answer▾ Answer
(i) They are equal; equal chords subtend equal angles. (ii) 54°. (iii) 54°; by the same theorem.

Frequently asked questions

What are the important theorems from the Circles chapter for Class 9 CBSE exams?

Key theorems include: the perpendicular from the center to a chord bisects the chord; the line joining the center to the midpoint of a chord is perpendicular to the chord; equal chords are equidistant from the center (and vice versa); and equal chords subtend equal angles at the center.

How do you find the length of a chord given its distance from the center?

Use the perpendicular from center to chord bisects the chord. Form a right triangle with the radius as hypotenuse, half the chord as one leg, and the distance as the other leg. Apply Pythagoras: (chord/2)² + distance² = radius², then solve for the chord length.

How is the circumradius of a right triangle determined?

In a right-angled triangle, the circumcenter is the midpoint of the hypotenuse. Thus, the hypotenuse is the diameter of the circumcircle, so the circumradius equals half the length of the hypotenuse.

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