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By Karan Singh Bisht
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Updated on 6 Oct 2025, 16:01 IST
CBSE Class 9 Chapter 9 Circles introduces you to the fascinating concepts of tangents, chords, and the unique properties that define a circle. You’ll dive into important theorems, such as the angle subtended by a chord at a point, and learn to tackle problems based on cyclic quadrilaterals.
This chapter plays a vital role in the CBSE Class 9 Maths Syllabus, as it strengthens your foundation for advanced studies. Mastering circles not only boosts your exam performance but also prepares you for higher-level geometry and beyond.
Consistent practice with Class 9 Maths Important Questions will sharpen your problem-solving skills, help you adapt to tricky exam patterns, and improve your chances of scoring full marks. With focused preparation, you’ll build confidence and develop a strong understanding of these essential geometrical concepts.
Before diving into the questions, it’s important to understand the key concepts:
Very Short Answer Type Questions: (1 Marks)
1. The distance between the center of a circle and a chord is called:
a) Radius
b) Diameter
c) Perpendicular distance
d) Secant
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Answer: c) Perpendicular distance
2. The line that intersects a circle at two distinct points is called:
a) Tangent
b) Secant
c) Chord
d) Diameter
Answer: b) Secant
3. A line touching a circle at only one point is called:
a) Tangent
b) Chord
c) Radius
d) Diameter
Answer: a) Tangent
4. The longest chord of a circle is:
a) Radius
b) Diameter
c) Secant
d) Tangent
Answer: b) Diameter
5. If two circles are equal, then their radii are:
a) Unequal
b) Equal
c) Double
d) Half
Answer: b) Equal
6. A tangent to a circle is always:
a) Perpendicular to the radius at the point of contact
b) Parallel to the radius
c) Equal to the diameter
d) None of these
Answer: a) Perpendicular to the radius at the point of contact
7. The angle subtended by a diameter of a circle at the circumference is:
a) 0°
b) 45°
c) 90°
d) 180°
Answer: c) 90°
8. The locus of points equidistant from a fixed point is a:
a) Line
b) Circle
c) Triangle
d) Square
Answer: b) Circle
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9. A quadrilateral is cyclic if the sum of opposite angles is:
a) 60°
b) 90°
c) 120°
d) 180°
Answer: d) 180°
10. In a circle, equal chords are equidistant from the:
a) Tangent
b) Circumference
c) Centre
d) Diameter
Answer: c) Centre
11. A line drawn from the center of a circle to the midpoint of a chord is:
a) Tangent
b) Secant
c) Perpendicular to the chord
d) Parallel to the chord
Answer: c) Perpendicular to the chord
12. The part of the circle bounded by an arc and the two radii is called:
a) Segment
b) Sector
c) Chord
d) Diameter
Answer: b) Sector
13. The line joining two points on the circle is known as:
a) Chord
b) Tangent
c) Radius
d) Arc
Answer: a) Chord
14. The region between a chord and the corresponding arc is called:
a) Sector
b) Segment
c) Tangent
d) Diameter
Answer: b) Segment
15. The measure of the angle subtended by an arc at the center is ______ the angle subtended at the circumference:
a) Equal to
b) Double
c) Half
d) One-fourth
Answer: b) Double
16. The perpendicular drawn from the center to a chord bisects the:
a) Circle
b) Radius
c) Chord
d) Tangent
Answer: c) Chord
17. If two tangents are drawn to a circle from an external point, then they are:
a) Unequal
b) Equal in length
c) Parallel
d) None of these
Answer: b) Equal in length
18. A circle can have how many tangents at a given point on it?
a) One
b) Two
c) Three
d) Infinite
Answer: a) One
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19. Which of the following is not a part of a circle?
a) Arc
b) Chord
c) Tangent
d) Diagonal
Answer: d) Diagonal
20. The angle in a semicircle is always:
a) 30°
b) 60°
c) 90°
d) 120°
Answer: c) 90°
Fill in the Blanks
Q. A line which intersects a circle at exactly one point is called a ________.
a) Chord
b) Tangent
c) Secant
d) Radius
Answer: b) Tangent
Q. The perpendicular drawn from the center of a circle to a chord ________ the chord.
a) Bisects
b) Doubles
c) Extends
d) Equals
Answer: a) Bisects
Q. The angle subtended by an arc at the center is ________ the angle subtended at any point on the circle.
a) Half
b) Twice
c) Equal to
d) One-third
Answer: b) Twice
Q. A quadrilateral is called a cyclic quadrilateral if all its ________ lie on a circle.
a) Angles
b) Vertices
c) Sides
d) Diagonals
Answer: b) Vertices
Q. The sum of the opposite angles of a cyclic quadrilateral is always ________.
a) 60°
b) 90°
c) 120°
d) 180°
Answer: d) 180°
Q. A line that intersects a circle at two points is called a ________.
a) Tangent
b) Secant
c) Radius
d) Diameter
Answer: b) Secant
Q. The radius drawn to the point of contact of a tangent is always ________ to the tangent.
a) Parallel
b) Perpendicular
c) Equal
d) Bisector
Answer: b) Perpendicular
Q. If two chords of a circle are equal in length, then they are ________ from the center.
a) Unequal
b) Equidistant
c) Parallel
d) Opposite
Answer: b) Equidistant
Write True or False:
Q. A tangent to a circle intersects the circle at exactly one point.
(A) True
(B) False
Answer: True
Q. Two tangents can be drawn from an external point to a circle.
(A) True
(B) False
Answer: True
Q. The perpendicular from the center of a circle to a chord always bisects the chord.
(A) True
(B) False
Answer: True
Q. The angle subtended by an arc at the center is equal to half the angle subtended at the circumference.
(A) True
(B) False
Answer: False
Q. All chords of a circle are equal in length.
(A) True
(B) False
Answer: False
Q. In a cyclic quadrilateral, the sum of opposite angles is always 180°.
(A) True
(B) False
Answer: True
Q. A line can intersect a circle at three distinct points.
(A) True
(B) False
Answer: False
Q. If two circles touch externally, the distance between their centers equals the sum of their radii.
(A) True
(B) False
Answer: True
Q. In the given figure, O is the centre of the circle with chords AP and BP being produced to R and Q respectively. If ∠QPR = 35°, find the measure of ∠AOB.
Given: O is the centre of the circle. Chords AP and BP are produced to R and Q respectively. Also, ∠QPR = 35°.
Step 1: Relate exterior and interior angles at P.
Since PR and PQ are straight extensions of PA and PB, ∠QPR is vertically opposite to the angle inside the circle at P, namely ∠APB.
∠APB = ∠QPR = 35°.
Step 2: Use the central–inscribed angle theorem.
The central angle ∠AOB and the inscribed angle ∠APB subtend the same arc AB. A central angle equals twice the corresponding inscribed angle.
∠AOB = 2 × ∠APB = 2 × 35° = 70°.
Final Answer: ∠AOB = 70°
Q. In the given figure, what is the measure of angle x ?
Solutions: We know that exterior angle of a cyclic quadrilateral is equal to interior opposite angle.
∠CBE = ∠ADC
x = 120°
Q. In the given figure, if O is the centre of circle and ∠POQ = 110°, then find ∠PRQ
Solutions: We know that angle subtended by an arc at the centre is double the angle subtended by it at the remaining part of the circle.
∠PRQ =1/2 ∠POQ
= 1/2 x 110°
= 55°
Q. In the given figure, ΔABC is an equilateral triangle and ABDC is a cyclic quadrilateral, then find the measure of ∠BDC.
Solutions: ΔABC is an equilateral triangle.
∠BAC = 60°.
Now,
∠BDC + ∠BAC = 180°
∠BDC + 60° = 180°
∠BDC = 180° − 60°
∠BDC = 120°
Q. In the given figure, O is the centre of the circle. PQ is a chord of the circle and R is any point on the circle. If ∠PRQ = l and ∠OPQ = m, then find l + m.
Solutions: ∠POQ = 2∠PRQ
∠POQ = 2l
In ΔPQO, OP = OQ = r
∠OQP = ∠OPQ = m
Also, ∠OPQ + ∠OQP + ∠POQ = 180°
m + m + 2l = 180°
2(l + m) = 180°
l + m = 90°
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Q. A circular park of radius 10 m is situated in a colony. Three students Ashok, Raman and Kanaihya are standing at equal distances on its circumference each having a toy telephone in his hands to talk each other about Honesty, Peace and Discipline.
(i) Find the length of the string of each phone.
(ii) Write the role of discipline in students’ life.
Solutions: Let us assume, A, B and C be the position of three students Ashok, Raman and Kanaihya respectively on the circumference of the circular park with centre O and radius 10 m. Since the centre of circle coincides with the centroid of the equilateral ΔABC.
ΔABC is an equilateral triangle.
∴ ∠BAC = 60°.
Now,
∠BDC + ∠BAC = 180°
∠BDC + 60° = 180°
∠BDC = 180° − 60°
∠BDC = 120°
∠POQ = 2∠PRQ
⇒ ∠POQ = 2l
In ΔPQO, OP = OQ = r
⇒ ∠OQP = ∠OPQ = m
Also,
∠OPQ + ∠OQP + ∠POQ = 180°
m + m + 2l = 180°
2(l + m) = 180°
l + m = 90°
Radius of circumscribed circle = 2/3 AD
10 = (2/3) AD ⇒ AD = 15 m
Now, AD ⊥ BC and let AB = BC = CA = x
BD = CD = (1/2)BC = x/2
In right ΔBDA, ∠D = 90°
AB2 = BD2 + AD2
x2 = (x2/4) + 225
x2 − (x2/4) = 225
x2 = 225 × (4/3) = 300
x = 10√3 m
Thus, the length of each string is 10√3 m.
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Important questions include proving theorems related to angles subtended by chords, properties of cyclic quadrilaterals, and solving problems involving tangents and secants.
You can find important questions with solutions on Infinity Learn, which offer comprehensive resources for Class 9 Maths Chapter 10.
To solve these questions, understand and apply circle theorems, practice constructing diagrams accurately, and refer to NCERT solutions for step-by-step guidance.
