NCERT Solutions for Class 9 Maths Chapter 9 Circles Ex 9.1

Exercise 9.1 

1. 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.

Ans: Given: Two given circles are congruent when they have the same radii. 

To Prove:  Equal chords of the congruent circles subtend equal angles at their centres.

A circle is a collection of equidistant points around a given point. This fixed location is known as the circle's centre, and this equal distance is known as the circle's radius. As a result, the radius of a circle determines its form. As a result, it can be seen that if two circles of equal radius are superimposed, both circles will cover each other. As a result, two circles are congruent if their radius is the same.

Consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths.

In $\vartriangle {\text{AOB}}$ and $\Delta {\text{C}}{{\text{O}}^\prime }{\text{D}}$,

${\text{AB}} = {\text{CD}}$ (Chords of same length) ${\text{OA}} = {{\text{O}}^\prime }{\text{C}}$ (Radii of congruent circles) ${\text{OB}} = {{\text{O}}^\prime }{\text{D}}$ (Radii of congruent circles)

$\therefore \Delta {\text{AOB}} \cong \Delta {\text{C}}{{\text{O}}^\prime }{\text{D}}$ (SAS congruence rule)

$ \Rightarrow \angle {\text{AOB}} = \angle {\text{C}}{{\text{O}}^\prime }{\text{D}}({\text{ByCPCT}})$

Hence, equal chords of congruent circles subtend equal angles at their centres.

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

Ans: Given: Chords of congruent circles subtend equal angles at their centres

To prove: Then the chords are equal

Let us consider two congruent circles (circles of the same radius) with centres as O and O'.

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In $\vartriangle {\text{AOB}}$ and $\Delta {\text{C}}{{\text{O}}^\prime }{\text{D}}$,

$\angle {\text{AOB}} = \angle {\text{C}}{{\text{O}}^\prime }{\text{D}}$ (Given)

${\text{OA}} = {{\text{O}}^\prime }{\text{C}}$ (Radii of congruent circles) ${\text{OB}} = {{\text{O}}^\prime }{\text{D}}$ (Radii of congruent circles)

$\therefore \Delta {\text{AOB}} \cong \Delta {\text{C}}{{\text{O}}^\prime }{\text{D}}$ (SAS congruence rule)

$ \Rightarrow {\text{AB}} = {\text{CD}}({\text{ByCPCT}})$ 

Hence, if the chords of congruent circles subtend equal angles at their respective centres, then the chords are equal.

NCERT Solutions for Class 9 Maths Chapter 9 Circles Exercise 9.1

Opting for the NCERT solutions for Ex 9.1 Class 9 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 9.1 Class 9 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 9 students who are thorough with all the concepts from the Maths textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 9 Maths Chapter 9 Exercise 9.1 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.

Besides these NCERT solutions for Class 9 Maths Chapter 9 Exercise 9.1, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it. 

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