NCERT Solution - Circles Class 9 Notes | EduRev

Class 9 : NCERT Solution - Circles Class 9 Notes | EduRev

 Page 1


Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 10 –Circles   Maths   
   
  
Exercise 10.1 Question 1:   
Fill in the blanks   
(i) The centre of a circle lies in __________ of the circle. (exterior/ interior) (ii) A point, 
whose distance from the centre of a circle is greater than its radius lies in __________ 
of the circle. (exterior/ interior)   
(iii) The longest chord of a circle is a __________ of the circle.   
(iv) An arc is a __________ when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and __________ of the circle.   
(vi) A circle divides the plane, on which it lies, in __________ parts.   
Answer:   
(i) The centre of a circle lies in interior of the circle.   
(ii) A point, whose distance from the centre of a circle is greater than its radius lies  
in exterior of the circle.   
(iii) The longest chord of a circle is a diameter of the circle.   
(iv) An arc is a semi-circle when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and chord of the circle.   
(vi) A circle divides the plane, on which it lies, in three parts.   
Question 2:   
Write True or False: Give reasons for your answers.   
(i) Line segment joining the centre to any point on the circle is a radius of the circle.   
Page 2


Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 10 –Circles   Maths   
   
  
Exercise 10.1 Question 1:   
Fill in the blanks   
(i) The centre of a circle lies in __________ of the circle. (exterior/ interior) (ii) A point, 
whose distance from the centre of a circle is greater than its radius lies in __________ 
of the circle. (exterior/ interior)   
(iii) The longest chord of a circle is a __________ of the circle.   
(iv) An arc is a __________ when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and __________ of the circle.   
(vi) A circle divides the plane, on which it lies, in __________ parts.   
Answer:   
(i) The centre of a circle lies in interior of the circle.   
(ii) A point, whose distance from the centre of a circle is greater than its radius lies  
in exterior of the circle.   
(iii) The longest chord of a circle is a diameter of the circle.   
(iv) An arc is a semi-circle when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and chord of the circle.   
(vi) A circle divides the plane, on which it lies, in three parts.   
Question 2:   
Write True or False: Give reasons for your answers.   
(i) Line segment joining the centre to any point on the circle is a radius of the circle.   
Cbse-spot.blogspot.com  
2 
  
(ii) A circle has only finite number of equal chords.   
(iii) If a circle is divided into three equal arcs, each is a major arc.   
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.   
(v) Sector is the region between the chord and its corresponding arc.   
(vi) A circle is a plane figure.   
Answer:   
(i) True. All the points on the circle are at equal distances from the centre of the 
circle, and this equal distance is called as radius of the circle.   
(ii) False. There are infinite points on a circle. Therefore, we can draw infinite 
number of chords of given length. Hence, a circle has infinite number of equal chords.   
(iii) False. Consider three arcs of same length as AB, BC, and CA. It can be observed 
that for minor arc BDC, CAB is a major arc. Therefore, AB, BC, and CA are minor arcs  
of the circle.   
   
(iv) True. Let AB be a chord which is twice as long as its radius. It can be observed 
that in this situation, our chord will be passing through the centre of the circle.   
Therefore, it will be the diameter of the circle.   
   
Page 3


Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 10 –Circles   Maths   
   
  
Exercise 10.1 Question 1:   
Fill in the blanks   
(i) The centre of a circle lies in __________ of the circle. (exterior/ interior) (ii) A point, 
whose distance from the centre of a circle is greater than its radius lies in __________ 
of the circle. (exterior/ interior)   
(iii) The longest chord of a circle is a __________ of the circle.   
(iv) An arc is a __________ when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and __________ of the circle.   
(vi) A circle divides the plane, on which it lies, in __________ parts.   
Answer:   
(i) The centre of a circle lies in interior of the circle.   
(ii) A point, whose distance from the centre of a circle is greater than its radius lies  
in exterior of the circle.   
(iii) The longest chord of a circle is a diameter of the circle.   
(iv) An arc is a semi-circle when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and chord of the circle.   
(vi) A circle divides the plane, on which it lies, in three parts.   
Question 2:   
Write True or False: Give reasons for your answers.   
(i) Line segment joining the centre to any point on the circle is a radius of the circle.   
Cbse-spot.blogspot.com  
2 
  
(ii) A circle has only finite number of equal chords.   
(iii) If a circle is divided into three equal arcs, each is a major arc.   
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.   
(v) Sector is the region between the chord and its corresponding arc.   
(vi) A circle is a plane figure.   
Answer:   
(i) True. All the points on the circle are at equal distances from the centre of the 
circle, and this equal distance is called as radius of the circle.   
(ii) False. There are infinite points on a circle. Therefore, we can draw infinite 
number of chords of given length. Hence, a circle has infinite number of equal chords.   
(iii) False. Consider three arcs of same length as AB, BC, and CA. It can be observed 
that for minor arc BDC, CAB is a major arc. Therefore, AB, BC, and CA are minor arcs  
of the circle.   
   
(iv) True. Let AB be a chord which is twice as long as its radius. It can be observed 
that in this situation, our chord will be passing through the centre of the circle.   
Therefore, it will be the diameter of the circle.   
   
Cbse-spot.blogspot.com  
3 
  
(v) False. Sector is the region between an arc and two radii joining the centre to the 
end points of the arc. For example, in the given figure, OAB is the sector of the  
circle.   
   
(vi) True. A circle is a two-dimensional figure and it can also be referred to as a plane 
figure.   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
Page 4


Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 10 –Circles   Maths   
   
  
Exercise 10.1 Question 1:   
Fill in the blanks   
(i) The centre of a circle lies in __________ of the circle. (exterior/ interior) (ii) A point, 
whose distance from the centre of a circle is greater than its radius lies in __________ 
of the circle. (exterior/ interior)   
(iii) The longest chord of a circle is a __________ of the circle.   
(iv) An arc is a __________ when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and __________ of the circle.   
(vi) A circle divides the plane, on which it lies, in __________ parts.   
Answer:   
(i) The centre of a circle lies in interior of the circle.   
(ii) A point, whose distance from the centre of a circle is greater than its radius lies  
in exterior of the circle.   
(iii) The longest chord of a circle is a diameter of the circle.   
(iv) An arc is a semi-circle when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and chord of the circle.   
(vi) A circle divides the plane, on which it lies, in three parts.   
Question 2:   
Write True or False: Give reasons for your answers.   
(i) Line segment joining the centre to any point on the circle is a radius of the circle.   
Cbse-spot.blogspot.com  
2 
  
(ii) A circle has only finite number of equal chords.   
(iii) If a circle is divided into three equal arcs, each is a major arc.   
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.   
(v) Sector is the region between the chord and its corresponding arc.   
(vi) A circle is a plane figure.   
Answer:   
(i) True. All the points on the circle are at equal distances from the centre of the 
circle, and this equal distance is called as radius of the circle.   
(ii) False. There are infinite points on a circle. Therefore, we can draw infinite 
number of chords of given length. Hence, a circle has infinite number of equal chords.   
(iii) False. Consider three arcs of same length as AB, BC, and CA. It can be observed 
that for minor arc BDC, CAB is a major arc. Therefore, AB, BC, and CA are minor arcs  
of the circle.   
   
(iv) True. Let AB be a chord which is twice as long as its radius. It can be observed 
that in this situation, our chord will be passing through the centre of the circle.   
Therefore, it will be the diameter of the circle.   
   
Cbse-spot.blogspot.com  
3 
  
(v) False. Sector is the region between an arc and two radii joining the centre to the 
end points of the arc. For example, in the given figure, OAB is the sector of the  
circle.   
   
(vi) True. A circle is a two-dimensional figure and it can also be referred to as a plane 
figure.   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
Cbse-spot.blogspot.com  
4 
  
   
   
   
   
   
   
   
   
   
   
   
   
   
Exercise 10.2 Question 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.   
Answer:   
A circle is a collection of points which are equidistant from a fixed point. This fixed point 
is called as the centre of the circle and this equal distance is called as radius of the 
circle. And thus, the shape of a circle depends on its radius. Therefore, it can be 
observed that if we try to superimpose two circles of equal radius, then both circles 
will cover each other. Therefore, two circles are congruent if they have equal radius. 
Consider two congruent circles having centre O and O' and two chords AB and CD of 
equal lengths.   
Page 5


Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 10 –Circles   Maths   
   
  
Exercise 10.1 Question 1:   
Fill in the blanks   
(i) The centre of a circle lies in __________ of the circle. (exterior/ interior) (ii) A point, 
whose distance from the centre of a circle is greater than its radius lies in __________ 
of the circle. (exterior/ interior)   
(iii) The longest chord of a circle is a __________ of the circle.   
(iv) An arc is a __________ when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and __________ of the circle.   
(vi) A circle divides the plane, on which it lies, in __________ parts.   
Answer:   
(i) The centre of a circle lies in interior of the circle.   
(ii) A point, whose distance from the centre of a circle is greater than its radius lies  
in exterior of the circle.   
(iii) The longest chord of a circle is a diameter of the circle.   
(iv) An arc is a semi-circle when its ends are the ends of a diameter.   
(v) Segment of a circle is the region between an arc and chord of the circle.   
(vi) A circle divides the plane, on which it lies, in three parts.   
Question 2:   
Write True or False: Give reasons for your answers.   
(i) Line segment joining the centre to any point on the circle is a radius of the circle.   
Cbse-spot.blogspot.com  
2 
  
(ii) A circle has only finite number of equal chords.   
(iii) If a circle is divided into three equal arcs, each is a major arc.   
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.   
(v) Sector is the region between the chord and its corresponding arc.   
(vi) A circle is a plane figure.   
Answer:   
(i) True. All the points on the circle are at equal distances from the centre of the 
circle, and this equal distance is called as radius of the circle.   
(ii) False. There are infinite points on a circle. Therefore, we can draw infinite 
number of chords of given length. Hence, a circle has infinite number of equal chords.   
(iii) False. Consider three arcs of same length as AB, BC, and CA. It can be observed 
that for minor arc BDC, CAB is a major arc. Therefore, AB, BC, and CA are minor arcs  
of the circle.   
   
(iv) True. Let AB be a chord which is twice as long as its radius. It can be observed 
that in this situation, our chord will be passing through the centre of the circle.   
Therefore, it will be the diameter of the circle.   
   
Cbse-spot.blogspot.com  
3 
  
(v) False. Sector is the region between an arc and two radii joining the centre to the 
end points of the arc. For example, in the given figure, OAB is the sector of the  
circle.   
   
(vi) True. A circle is a two-dimensional figure and it can also be referred to as a plane 
figure.   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
Cbse-spot.blogspot.com  
4 
  
   
   
   
   
   
   
   
   
   
   
   
   
   
Exercise 10.2 Question 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.   
Answer:   
A circle is a collection of points which are equidistant from a fixed point. This fixed point 
is called as the centre of the circle and this equal distance is called as radius of the 
circle. And thus, the shape of a circle depends on its radius. Therefore, it can be 
observed that if we try to superimpose two circles of equal radius, then both circles 
will cover each other. Therefore, two circles are congruent if they have equal radius. 
Consider two congruent circles having centre O and O' and two chords AB and CD of 
equal lengths.   
Cbse-spot.blogspot.com  
5 
  
   
In ?AOB and ?CO'D,   
AB = CD (Chords of same length)   
OA = O'C (Radii of congruent circles)   
OB = O'D (Radii of congruent circles)   
  
 ?AOB  ?CO'D (SSS congruence rule)   
 
AOB = CO'D (By CPCT)   
Hence, equal chords of congruent circles subtend equal angles at their centres.   
Question 2:   
Prove that if chords of congruent circles subtend equal angles at their centres, then the 
chords are equal.   
   
Answer:   
Let us consider two congruent circles (circles of same radius) with centres as O and O'.   
   
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