# NCERT Solutions for Class 9 Maths Chapter 10

NCERT Solutions for Class 9 Maths Chapter 10 Circles (वृत) Exercise 10.1, Exercise 10.2, Exercise 10.3, Exercise 10.4, Exercise 10.5 and Exercise 10.6 in English Medium and Hindi Medium updated for session 2020-21. UP Board Solutions for Class 9 Maths Chapter 10 Prashnavali 10.1, Prashnavali 10.2, Prashnavali 10.3, Prashnavali 10.4, Prashnavali 10.5 and Prashnavali 10.6 in Hindi Medium to study online or download in PDF format free for the new CBSE session 2020-2021. All the solutions for 9th Maths are updated for the new session based on NCERT Books 2020-21. Most of the state board like UP Board, MP Board, Jammu and Kashmir, Uttarakhand, etc.,

, are now following NCERT Textbooks for their academic session. So, these solutions are applicable for all these boards also. All the questions are explained well using the theorems of circles and giving proper examples. In few questions some axioms of circles are also used as theorems.## NCERT Solutions for Class 9 Maths Chapter 10

Class: | 9 |

Subject: | Maths – गणित |

Chapter 10: | Circles |

### 9th Maths Chapter 10 Solutions in English & Hindi Medium

NCERT Solutions 2020-21 and CBSE Offline Apps of other subjects are also available free to use or download. Solutions are prepared for CBSE and UP Board students appearing in 2020-21 CBSE Exams and following the updated NCERT Books. UP Board NCERT Solutions for Class 9 Maths Chapter 10 Circles all Exercises are given below.

### 9th Maths Exercise 10.1 Solutions

### 9th Maths Exercise 10.2 Solutions

### 9th Maths Exercise 10.3 Solutions

### 9th Maths Exercise 10.4 Solutions

### 9th Maths Exercise 10.5 Solutions

### 9th Maths Exercise 10.6 Solutions

### Study Material for 2020-21

#### Class 9 Maths Exercise 10.1 and 10.2 Solutions in Video

#### Class 9 Maths Exercise 10.3 and 10.4 Solutions in Video

#### Class 9 Maths Exercise 10.5 and 10.6 Solutions in Video

##### What do understand by a circle?

Circle: A circle is the collection of all points in a plane, which are at a fixed distance from a fixed point in the plane.

##### What are the components of a circle?

1. Diameter: It is the longest chord of the circle.

2. Circumference: The length of complete circle is called its circumference.

3. Arc: A piece of circle between two point is called arc.

4. Segment: The region between a chord and either of its arcs is called a segment of circular region.

##### What are the main Properties related to a circle?

Properties related to CIRCLE

1. Equal chords of a circle subtend equal angles at the centre.

2. If the angles subtended by two chords of a circle at the centre are equal, the chords are also equal.

3. The perpendicular from the centre of a circle to a chord bisects the chord.

4. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

5. There is one and only one circle passing through three non-collinear points.

#### Important Theorems on Circles Class 9 Maths Chapter 10

- Equal chords of a circle are equidistant from the centre and cords equidistant from the centre of a circle are equal.
- If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs are congruent.
- Congruent arcs of a circle subtend equal angles at the centre.
- The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal.
- Angle in a semi-circle is a right angle.
- The sum of either pair of opposite angles of a cyclic quadrilateral is 180 and if the sum of a pair of opposite angles of a quadrilateral is 180, then the quadrilateral is cyclic.

### Important Questions on 9th Maths Chapter 10

सिद्ध करना है: ∠BAC = ∠QPR

उपपत्ति: ABC और PQR में,

BC = QR [∵ दिया है]

AB = PQ [∵ सर्वांगसम वृत्तों की त्रिज्याएँ]

AC = PR [∵ सर्वांगसम वृत्तों की त्रिज्याएँ]

अतः, ABC ≅ PQR [∵SSS सर्वांगसमता नियम]

∠BAC = ∠QPR [∵ सर्वांगसम त्रिभुज के संगत भाग बराबर होते हैं]

Because, there are infinite number of equal chords in a circle.

False.

Because, between chord and arc a segment is formed. Sector is the region which is formed between radii and arc.

सिद्ध करना है: AB को व्यास मानकर खींचा गया वृत्त विकर्णों के प्रतिच्छेद बिन्दु O से होकर जाता है।

उपपत्ति: ABCD एक समचतुर्भु है। अतः, ∠AOC = 90°

[∵ समचतुर्भुज के विकर्ण एक दूसरे को लम्ब समद्विभाजित करते हैं]

AB को व्यास मानकर खींचा गया वृत्त बिन्दु O से हो कर जाएगा।

[∵ अर्धवृत्त में बना कोण समकोण होता है]

अतः, समचतुर्भुज की किसी भुजा को व्यास मानकर खींचा गया वृत्त उसके विकर्णों के प्रतिच्छेद बिन्दु से होकर जाता है।

Hence, ∠ADC = 90° and ∠ABC = 90° … (1)

[∵ Angle in a semicircle is a right angle.]

Similarly, BD is diameter of circle.

Hence, ∠BAD = 90° and ∠BCD = 90° … (2)

[∵ Angle in a semicircle is a right angle.]

From the equation (1) and (2),

∠ADC =∠ABC = ∠BAD = ∠BCD = 90°

Hence, ABCD is a rectangle.

To prove: ∠CAD = ∠CBD.

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

Hence, points A, B, C and D lie on the same circle.

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

Therefore,

∠CAD = ∠CBD

[∵ Angles in the same segments are equal]