NCERT Solutions for Class 9 Maths Chapter 9 Exercise 9.1

Team Careers360Updated on 03 May 2025, 03:14 PM IST

A circle is a two-dimensional geometrical figure in which all points on the surface of the circle are equidistant from the centre point. A circle can be characterised as a shut (closed), two-dimensional curved shape, with no pointed vertex, with zero corners, such that every point of the circle is equidistant from the fixed centre of the circle. Radius is the distance from the centre of the circle to any point on the circumference of the circle. A circle's chord is the line segment that connects two locations on its circumference. The circumference of a circle is the perimeter of a circle; we can divide the circumference into two parts: a major arc and a minor arc. The major arc includes the majority portion of the arc, and the minor arc includes the minority portion of the arc. You will see that the angles subtended by two or more equal chords of a circle at the centre are equal if you draw two or more equal chords of a circle and determine the angles they occupy in the centre.

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NCERT Solutions Class 9 Maths Chapter 1: Exercise 9.1
Topics Covered in Chapter 9, Circles: Exercise 9.1
NCERT Solutions of Class 9 Subject Wise
Subject-Wise NCERT Exemplar Solutions

Students can also get all NCERT solutions according to the standard and subjects. In Class 9 Maths chapter 9 exercise 9.1, we will cover the theorems of the angle subtended by a chord at a point in depth. The following exercises are also included in the NCERT Book Class 9 Mathematics chapter 9 exercise 9.1. Class 9 Maths chapter 9 exercise 9.1 is about the angle subtended by a chord at a point.

NCERT Solutions Class 9 Maths Chapter 1: Exercise 9.1

Q1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

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In the given figure, two congruent circles are given and have centers O and O', and chords are PQ and XY, respectively.
In Δ POQ and XO'Y
PQ = XY (Given)
OP = O'X (Radius of congruent triangle)
OQ = O'Y (Radius of congruent triangle)
ΔPOQ ≅ ΔXO'Y (By SSS rule)
Therefore, ∠POQ = ∠XO'Y (By CPCT)

Q2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Solution:
Given: Two congruent circles have equal angles.
To Prove: Two congruent circles have equal angles, and then the chords are equal.

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In Δ POQ and XO'Y
∠POQ = ∠XO'Y (Given)
OP = O'X (Radius of congruent triangle)
OQ = O'Y (Radius of congruent triangle)
ΔPOQ ≅ ΔXO'Y (By SSS rule)
Therefore, PQ = XY (By CPCT)

Also Read

Topics Covered in Chapter 9, Circles: Exercise 9.1

As we know NCERT solutions for exercise 9.1 Class 9 Maths chapter 9 are about the angle subtended by a chord at a point. But along with the topics about circles, such as chords, arc, major arc, minor arc, semicircle, we have also used the information about the congruence of the triangle, which we have covered in Class 9 Chapter 7 triangle. Some of the rules and properties which we are using are the side-side-side rule, the side-angle-side rule, and there are some theorems which are covered before Class 9 Maths chapter 9 exercise 9.1. These theorems are at the centre of a circle, equal chords subtend equal angles. If the angles subtended by the chords of a circle at its centre are equal, then the chords are equal, as well as the theorem.

Also See

NCERT Solutions of Class 9 Subject Wise

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

Subject-Wise NCERT Exemplar Solutions

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

Frequently Asked Questions (FAQs)

Q: What is the core concept of NCERT solutions for Class 9 Maths exercise 10.1?

This exercise deals with all the difficult questions regarding circles and its properties.

Q: Define a circle.

A circle can be defined as a closed, two-dimensional curved shape, with zero corners such that every point of the circle is equidistant from the fixed centre of the circle.

Q: What is the radius of a circle?

The radius of a circle is the straight line extending from the centre of a circle to the circumference or surface.

Q: What is the relation between radius and diameter of a circle?

diameter=2 radius

Q: Find the diameter of the circle if the radius is 10 cm?

The relation between the radius and diameter of a circle is:

diameter=2 radius

diameter=20 cm

Q: Which segment contains the centre of the circle?

The major segment of the circle contains the centre of the circle whereas the minor segment doesn’t include the centre.

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