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NCERT Solutions for Class 9 Maths Chapter 9 Exercise 9.3 - Circles

NCERT Solutions for Class 9 Maths Chapter 9 Exercise 9.3 - Circles

Edited By Vishal kumar | Updated on May 03, 2025 03:16 PM IST

The smooth curve joining two points on a circle is called an arc. It is part of the circumference of the circle. In a circle chord is a line segment that joins the two endpoints on the circumference. This chord divides the circle into two different parts: one is a major part, and it is called the major arc, and the other is a minor part, and it is called the minor arc. If two chords are equal in a circle, then their corresponding arcs are congruent.

This Story also Contains
  1. NCERT Solutions for Class 9 Maths Chapter 9 – Circles Exercise 9.3
  2. Access Circles Class 9 Chapter 9 Exercise: 9.3
  3. Topics Covered in Chapter 9 Circles: Exercise 9.3
  4. NCERT Solutions of Class 9 Subject Wise
  5. NCERT Subject-Wise Exemplar Solutions
NCERT Solutions for Class 9 Maths Chapter 9 Exercise 9.3 - Circles
NCERT Solutions for Class 9 Maths Chapter 9 Exercise 9.3 - Circles

Chapter 9 in class 9 is about circles and the basic definition of circles, chords, segments, arcs and related questions with solutions already discussed in exercises 9.1 and 9.2. In exercise 9.3, students get the solutions to the questions given in the NCERT exercise. These questions are based on the chords, arcs, and perpendicular lines. These solutions will help you to understand the concept. All the solutions are detailed and explained in understandable language. This will help you to ace the topic through regular practice of the questions. Students can also find solutions to the NCERT Books in PDF format, so the solutions can be used offline. Students can also download NCERT Solutions according to their standard and requirements.

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**Please be aware that this chapter has been renumbered as Chapter 9 in the CBSE Syllabus for the academic year 2023-24.

NCERT Solutions for Class 9 Maths Chapter 9 – Circles Exercise 9.3

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Access Circles Class 9 Chapter 9 Exercise: 9.3

Q1 In Fig. 10.36 , A,B and C are three points on a circle with centre O such that ∠BOC=30∘ and ∠AOB=60∘ . If D is a point on the circle other than the arc ABC, find ∠ADC .

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Solution:
∠ AOC = ∠ AOB + ∠ BOC = 60∘+30∘=90∘

∠ AOC = 2 ∠ ADC (angle subtended by an arc at the centre is double the angle subtended by it at any)

∠ADC=12∠AOC

⇒∠ADC=1290∘=45∘

Q2 A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Solution:
Given: A chord of a circle is equal to the radius of the circle i.e. OA = AB.
To find: ∠ADB and ∠ ACB.

1745297632182

In △ OAB,OA = AB (Given)

OA = OB (Radii of circle)

So, OA = OB = AB

⇒ AOB is an equilateral triangle.

So, ∠ AOB = 60∘

∠ AOB = 2 ∠ ADB

⇒∠ADB=12∠AOB

⇒∠ADB=1260∘=30

ACBD is a cyclic quadrilateral.

So, ∠ ACB+ ∠ ADB = 180∘

⇒∠ACB+30∘=180∘

⇒∠ACB=180∘−30∘=150∘

Q3 In Fig. 10.37 , ∠PQR=100∘ , where P, Q and R are points on a circle with centre O. Find ∠OPR .

1745321888736

Solution:

1745321929523

Construction: Join PO and OR.
PQSR is a cyclic quadrilateral.

So, ∠ PSR + ∠ PQR = 180∘

⇒∠PSR+100∘=180∘

⇒∠PSR=180∘−100∘=80∘

Here, ∠ POR = 2 ∠ PSR

⇒∠POR=2×80∘=160∘

In △ OPR,

OP = OR (Radii)

∠ ORP = ∠ OPR (the angles opposite to equal sides)

In △ OPR,

∠ OPR+ ∠ ORP + ∠ POR = 180∘

⇒2∠OPR+160∘=180∘

⇒2∠OPR=180∘−160∘

⇒2∠OPR=20∘

⇒∠OPR=10∘

Q4 In Fig. 10.38 , ∠ABC=69∘,∠ACB=31∘, find ∠BDC


1745297886746

Solution:

In △ ABC,

∠ A+ ∠ ABC+ ∠ ACB= 180∘

⇒∠A+69∘+31∘=180∘

⇒∠A+100∘=180∘

⇒∠A=180∘−100∘

⇒∠A=80∘

∠ A = ∠ BDC = 80∘ (Angles in same segment)

Q5 In Fig. 10.39 , A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC=130∘ and ∠ECD=20∘ . Find ∠BAC

1745297908720 .

Solution:

∠ DEC+ ∠ BEC = 180∘ (linear pairs)

⇒ ∠ DEC+ 130∘ = 180∘ ( ∠ BEC = 130∘ )

⇒ ∠ DEC = 180∘ - 130∘

⇒ ∠ DEC = 50∘

In △ DEC,

∠ D+ ∠ DEC+ ∠ DCE = 180∘

⇒∠D+50∘+20∘=180∘

⇒∠D+70∘=180∘

⇒∠D=180∘−70∘=110∘

∠ D = ∠ BAC (angles in same segment are equal )

∠ BAC = 110∘

Q6 ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC=70∘ , ∠BAC is 30∘ , find ∠BCD . Further, if AB=BC , find ∠ECD .

Solution:
1745297946422

∠BDC=∠BAC (angles in the same segment are equal )

∠BDC=30∘

In △BDC,

∠BCD+∠BDC+∠DBC=180∘

⇒∠BCD+30∘+70∘=180∘

⇒∠BCD+100∘=180∘

⇒∠BCD=180∘−100∘=80∘

If AB = BC ,then

∠BCA=∠BAC

⇒∠BCA=30∘

Here, ∠ECD+∠BCE=∠BCD

⇒∠ECD+30∘=80∘

⇒∠ECD=80∘−30∘=50∘


Q7 If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Solution:
1745298019285

AC is the diameter of the circle.

Thus, ∠ADC=90∘ and ∠ABC=90∘ ............................1(Angle in a semi-circle is a right angle)

Similarly, BD is the diameter of the circle.

Thus, ∠BAD=90∘ and ∠BCD=90∘ ............................2(Angle in a semi-circle is a right angle)

From 1 and 2, we get

∠BCD=∠ADC=∠ABC=∠BAD=90∘

Hence, ABCD is a rectangle.

Q8 If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Solution:

1745298063019

Given: ABCD is a trapezium.

Construction: Draw AD || BE.

Proof: In quadrilateral ABED,

AB || DE (Given )

AD || BE ( By construction )

Thus, ABED is a parallelogram.

AD = BE (Opposite sides of parallelogram )

AD = BC (Given )

so, BE = BC

In △ EBC,

BE = BC (Proved above )

Thus, ∠C=∠2 ...........1(angles opposite to equal sides )

∠A=∠1 ...............2(Opposite angles of the parallelogram )

From 1 and 2, we get

∠1+∠2=180∘ (linear pair)

⇒∠A+∠C=180∘

Thus, ABED is a cyclic quadrilateral.


Q9 Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 10.40 ). Prove that ∠ACP=∠QCD .

1745298090691

Solution:

∠ABP=∠QBD ................1(vertically opposite angles)

∠ACP=∠ABP ..................2(Angles in the same segment are equal)

∠QBD=∠QCD .................3(angles in the same segment are equal)

From 1,2,3 ,we get

∠ACP=∠QCD

Q10 If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Answer:

Given: circles are drawn taking two sides of a triangle as diameters.

Construction: Join AD.

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Proof: AB is the diameter of the circle and ∠ ADB is formed in a semi-circle.

∠ ADB = 90∘ ........................1(angle in a semi-circle)

Similarly,

AC is the diameter of the circle and ∠ ADC is formed in a semi-circle.

∠ ADC = 90∘ ........................2(angle in a semi-circle)

From 1 and 2, we have

∠ ADB+ ∠ ADC= 90∘ + 90∘ = 180∘

∠ ADB and ∠ ADC are forming a linear pair. So, BDC is a straight line.

Hence, point D lies on this side.

Q11 ABC and ADC are two right triangles with common hypotenuse AC. Prove that
∠CAD=∠CBD .

Solution:

Given: ABC and ADC are two right triangles with common hypotenuse AC.

To prove : ∠CAD=∠CBD

Proof :

1745298195372

Triangle ABC and ADC are on common base BC and ∠ BAC = ∠ BDC.

Thus, point A, B, C, D lie in the same circle.

(If a line segment joining two points subtends equal angles at two other points lying on the same side of line containing line segment, four points lie on the circle.)

∠ CAD = ∠ CBD (Angles in the same segment are equal)


Q12 Prove that a cyclic parallelogram is a rectangle.
Solution:

Given: ABCD is a cyclic quadrilateral.

To prove: ABCD is a rectangle.

Proof :

1745298231341

In cyclic quadrilateral ABCD.

∠A+∠C=180∘ .......................1(sum of either pair of opposite angles of a cyclic quadrilateral)

∠A=∠C ........................................2(opposite angles of a parallelogram are equal )

From 1 and 2,

∠A+∠A=180∘

⇒2∠A=180∘

⇒∠A=90∘

We know that a parallelogram with one angle a right angle is a rectangle.

Hence, ABCD is a rectangle.


Also Read:

Topics Covered in Chapter 9 Circles: Exercise 9.3

1. This exercise 9.3 class 9 maths includes some important points like when two circle chords are equal, their corresponding arcs are congruent, and when two arcs are congruent, their corresponding chords are equal.
2. A circle's congruent arcs (or equal arcs) subtend equal angles at the centre.
3. The angle of an arc at its centre is double that of any point on the circle's remainder.
4. If all four vertices of a quadrilateral ABCD lie on a circle, it is called cyclic.
5. The sum of any pair of opposite angles of a cyclic quadrilateral is 180°.
6. If a quadrilateral's sum of opposite angles is 180°, the quadrilateral is cyclic.

Also, See

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NCERT Solutions of Class 9 Subject Wise

Students must check the NCERT solutions for Class 9 Maths and Science given below:

NCERT Subject-Wise Exemplar Solutions

Students must check the NCERT exemplar solutions for Class 9 Maths and Science given below:

Frequently Asked Questions (FAQs)

1. Define cyclic quadrilateral?

The quadrilateral with all of its vertices on the same circle is known as a cyclic quadrilateral.

2. What are the theorems for cyclic quadrilaterals?

If the sum of opposite angles in a quadrilateral is 180 degrees, the quadrilateral is said to be cyclic.

3. What is the angle subtended by diameter in the semicircle?

Diameter subtends right angle.

4. When a circle is drawn, it divides the plane into how many parts?

A circle divides the plane into 3 parts.

5. What is the condition for two circles to be congruent?

Circles should have the same radius.

6. What is the number of circles that can pass through 3 non-collinear points?

Only one circle can pass through 3 noncollinear points.

7. What is the relation between the diameter and radius?

The diameter is double the radius.

8. Define segment in circle?

A segment is defined as the region between an arc and the chord of the circle.

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