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Page 1
Q u e s t i o n : 3
The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.
S o l u t i o n :
Let AB be a chord of a circle with centre O and radius 8 cm such that
AB = 12 cm
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
Hence the distance of chord from the centre .
Q u e s t i o n : 4
Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.
S o l u t i o n :
Given that OA = 10 cm and OL = 5 cm, we have to find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 10 cm such that AO = 10 cm
We draw and join OA.
Since, the perpendiculars from the centre of a circle to a chord bisect the chord.
Now in we have
Page 2
Q u e s t i o n : 3
The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.
S o l u t i o n :
Let AB be a chord of a circle with centre O and radius 8 cm such that
AB = 12 cm
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
Hence the distance of chord from the centre .
Q u e s t i o n : 4
Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.
S o l u t i o n :
Given that OA = 10 cm and OL = 5 cm, we have to find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 10 cm such that AO = 10 cm
We draw and join OA.
Since, the perpendiculars from the centre of a circle to a chord bisect the chord.
Now in we have
Hence the length of chord
Q u e s t i o n : 5
Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.
S o l u t i o n :
Given that and , find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 6 cm such that
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
? AL = v 20 = 4. 47
AB = 2 ×AL = 2 ×4. 47 = 8. 94 cm
Hence the length of the chord is 8.94 cm.
Q u e s t i o n : 6
Give a method to find the centre of a given circle.
S o l u t i o n :
Let A, B and C are three distinct points on a circle .
Now join AB and BC and draw their perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the centre of given circle.
Hence O is the centre of circle .
Q u e s t i o n : 7
Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.
S o l u t i o n :
Let MN is the diameter and chord AB of circle C(O, r) then according to the question
AP = BP.
Then we have to prove that .
Join OA and OB.
In ?AOP and ?BOP
Radiiofthesamecircle
Page 3
Q u e s t i o n : 3
The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.
S o l u t i o n :
Let AB be a chord of a circle with centre O and radius 8 cm such that
AB = 12 cm
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
Hence the distance of chord from the centre .
Q u e s t i o n : 4
Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.
S o l u t i o n :
Given that OA = 10 cm and OL = 5 cm, we have to find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 10 cm such that AO = 10 cm
We draw and join OA.
Since, the perpendiculars from the centre of a circle to a chord bisect the chord.
Now in we have
Hence the length of chord
Q u e s t i o n : 5
Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.
S o l u t i o n :
Given that and , find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 6 cm such that
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
? AL = v 20 = 4. 47
AB = 2 ×AL = 2 ×4. 47 = 8. 94 cm
Hence the length of the chord is 8.94 cm.
Q u e s t i o n : 6
Give a method to find the centre of a given circle.
S o l u t i o n :
Let A, B and C are three distinct points on a circle .
Now join AB and BC and draw their perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the centre of given circle.
Hence O is the centre of circle .
Q u e s t i o n : 7
Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.
S o l u t i o n :
Let MN is the diameter and chord AB of circle C(O, r) then according to the question
AP = BP.
Then we have to prove that .
Join OA and OB.
In ?AOP and ?BOP
Radiiofthesamecircle
AP = BP (P is the mid point of chord AB)
OP = OP Common
Therefore,
bycpct
Hence, proved.
Q u e s t i o n : 8
A line segment AB is of length 5cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your
answer.
S o l u t i o n :
Given that a line AB = 5 cm, one circle having radius of which is passing through point A and B and other circle of radius .
As we know that the largest chord of any circle is equal to the diameter of that circle.
So,
There is no possibility to draw a circle whose diameter is smaller than the length of the chord.
Q u e s t i o n : 9
An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.
S o l u t i o n :
Let ABC be an equilateral triangle of side 9 cm and let AD be one of its medians. Let G be the centroid of . Then
We know that in an equilateral triangle centroid coincides with the circumcentre. Therefore, G is the centre of the circumcircle with circumradius GA.
As per theorem, G is the centre and . Therefore,
In we have
Therefore radius AG =
2
3
AD = 3v 3 cm
Q u e s t i o n : 1 0
Given an arc of a circle, complete the circle.
S o l u t i o n :
Let PQ be an arc of the circle.
In order to complete the circle. First of all we have to find out its centre and radius.
Now take a point R on the arc PQ and join PR and QR.
Draw the perpendicular bisectors of PR and QR respectively.
Let these perpendicular bisectors intersect at point O.
Then OP = OQ, draw a circle with centre O and radius OP = OQ to get the required circle.
Q u e s t i o n : 1 1
Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
S o l u t i o n :
Given that two different pairs of circles in the figure.
Page 4
Q u e s t i o n : 3
The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.
S o l u t i o n :
Let AB be a chord of a circle with centre O and radius 8 cm such that
AB = 12 cm
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
Hence the distance of chord from the centre .
Q u e s t i o n : 4
Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.
S o l u t i o n :
Given that OA = 10 cm and OL = 5 cm, we have to find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 10 cm such that AO = 10 cm
We draw and join OA.
Since, the perpendiculars from the centre of a circle to a chord bisect the chord.
Now in we have
Hence the length of chord
Q u e s t i o n : 5
Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.
S o l u t i o n :
Given that and , find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 6 cm such that
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
? AL = v 20 = 4. 47
AB = 2 ×AL = 2 ×4. 47 = 8. 94 cm
Hence the length of the chord is 8.94 cm.
Q u e s t i o n : 6
Give a method to find the centre of a given circle.
S o l u t i o n :
Let A, B and C are three distinct points on a circle .
Now join AB and BC and draw their perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the centre of given circle.
Hence O is the centre of circle .
Q u e s t i o n : 7
Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.
S o l u t i o n :
Let MN is the diameter and chord AB of circle C(O, r) then according to the question
AP = BP.
Then we have to prove that .
Join OA and OB.
In ?AOP and ?BOP
Radiiofthesamecircle
AP = BP (P is the mid point of chord AB)
OP = OP Common
Therefore,
bycpct
Hence, proved.
Q u e s t i o n : 8
A line segment AB is of length 5cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your
answer.
S o l u t i o n :
Given that a line AB = 5 cm, one circle having radius of which is passing through point A and B and other circle of radius .
As we know that the largest chord of any circle is equal to the diameter of that circle.
So,
There is no possibility to draw a circle whose diameter is smaller than the length of the chord.
Q u e s t i o n : 9
An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.
S o l u t i o n :
Let ABC be an equilateral triangle of side 9 cm and let AD be one of its medians. Let G be the centroid of . Then
We know that in an equilateral triangle centroid coincides with the circumcentre. Therefore, G is the centre of the circumcircle with circumradius GA.
As per theorem, G is the centre and . Therefore,
In we have
Therefore radius AG =
2
3
AD = 3v 3 cm
Q u e s t i o n : 1 0
Given an arc of a circle, complete the circle.
S o l u t i o n :
Let PQ be an arc of the circle.
In order to complete the circle. First of all we have to find out its centre and radius.
Now take a point R on the arc PQ and join PR and QR.
Draw the perpendicular bisectors of PR and QR respectively.
Let these perpendicular bisectors intersect at point O.
Then OP = OQ, draw a circle with centre O and radius OP = OQ to get the required circle.
Q u e s t i o n : 1 1
Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
S o l u t i o n :
Given that two different pairs of circles in the figure.
As we see that only two points A, B of first pair of circle and C, D of the second pair of circles are common points.
Thus only two points are common in each pair of circle.
Q u e s t i o n : 1 2
Suppose you are given a circle. Give a construction to find its centre.
S o l u t i o n :
Given a circle C(O, r).
We take three points A, B and C on the circle.
Join AB and BC.
Draw the perpendicular bisector of chord AB and BC.
Let these bisectors intersect at point O.
Hence, O is the centre of circle.
Q u e s t i o n : 1 3
The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?
S o l u t i o n :
Let AB and CD be two parallel chord of the circle with centre O such that AB = 6 cm, CD = 8 cm and OP = 4 cm. let the radius of the circle be cm.
According to the question, we have to find OQ
Draw and as well as point O, Q, and P are collinear.
Let
Join OA and OC, then
OA = OC = r
Now and
So, AP = 3 cm and CQ = 4 cm
In we have
And in
Q u e s t i o n : 1 4
Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel, If the distance between AB and CD is 3 cm, find the radius of the circle.
S o l u t i o n :
Let AB and CD be two parallel chord of the circle with centre O such that AB = 5 cm and CD = 11 cm. let the radius of the circle be cm.
Page 5
Q u e s t i o n : 3
The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.
S o l u t i o n :
Let AB be a chord of a circle with centre O and radius 8 cm such that
AB = 12 cm
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
Hence the distance of chord from the centre .
Q u e s t i o n : 4
Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.
S o l u t i o n :
Given that OA = 10 cm and OL = 5 cm, we have to find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 10 cm such that AO = 10 cm
We draw and join OA.
Since, the perpendiculars from the centre of a circle to a chord bisect the chord.
Now in we have
Hence the length of chord
Q u e s t i o n : 5
Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.
S o l u t i o n :
Given that and , find the length of chord AB.
Let AB be a chord of a circle with centre O and radius 6 cm such that
We draw and join OA.
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Now in we have
? AL = v 20 = 4. 47
AB = 2 ×AL = 2 ×4. 47 = 8. 94 cm
Hence the length of the chord is 8.94 cm.
Q u e s t i o n : 6
Give a method to find the centre of a given circle.
S o l u t i o n :
Let A, B and C are three distinct points on a circle .
Now join AB and BC and draw their perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the centre of given circle.
Hence O is the centre of circle .
Q u e s t i o n : 7
Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.
S o l u t i o n :
Let MN is the diameter and chord AB of circle C(O, r) then according to the question
AP = BP.
Then we have to prove that .
Join OA and OB.
In ?AOP and ?BOP
Radiiofthesamecircle
AP = BP (P is the mid point of chord AB)
OP = OP Common
Therefore,
bycpct
Hence, proved.
Q u e s t i o n : 8
A line segment AB is of length 5cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your
answer.
S o l u t i o n :
Given that a line AB = 5 cm, one circle having radius of which is passing through point A and B and other circle of radius .
As we know that the largest chord of any circle is equal to the diameter of that circle.
So,
There is no possibility to draw a circle whose diameter is smaller than the length of the chord.
Q u e s t i o n : 9
An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.
S o l u t i o n :
Let ABC be an equilateral triangle of side 9 cm and let AD be one of its medians. Let G be the centroid of . Then
We know that in an equilateral triangle centroid coincides with the circumcentre. Therefore, G is the centre of the circumcircle with circumradius GA.
As per theorem, G is the centre and . Therefore,
In we have
Therefore radius AG =
2
3
AD = 3v 3 cm
Q u e s t i o n : 1 0
Given an arc of a circle, complete the circle.
S o l u t i o n :
Let PQ be an arc of the circle.
In order to complete the circle. First of all we have to find out its centre and radius.
Now take a point R on the arc PQ and join PR and QR.
Draw the perpendicular bisectors of PR and QR respectively.
Let these perpendicular bisectors intersect at point O.
Then OP = OQ, draw a circle with centre O and radius OP = OQ to get the required circle.
Q u e s t i o n : 1 1
Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
S o l u t i o n :
Given that two different pairs of circles in the figure.
As we see that only two points A, B of first pair of circle and C, D of the second pair of circles are common points.
Thus only two points are common in each pair of circle.
Q u e s t i o n : 1 2
Suppose you are given a circle. Give a construction to find its centre.
S o l u t i o n :
Given a circle C(O, r).
We take three points A, B and C on the circle.
Join AB and BC.
Draw the perpendicular bisector of chord AB and BC.
Let these bisectors intersect at point O.
Hence, O is the centre of circle.
Q u e s t i o n : 1 3
The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?
S o l u t i o n :
Let AB and CD be two parallel chord of the circle with centre O such that AB = 6 cm, CD = 8 cm and OP = 4 cm. let the radius of the circle be cm.
According to the question, we have to find OQ
Draw and as well as point O, Q, and P are collinear.
Let
Join OA and OC, then
OA = OC = r
Now and
So, AP = 3 cm and CQ = 4 cm
In we have
And in
Q u e s t i o n : 1 4
Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel, If the distance between AB and CD is 3 cm, find the radius of the circle.
S o l u t i o n :
Let AB and CD be two parallel chord of the circle with centre O such that AB = 5 cm and CD = 11 cm. let the radius of the circle be cm.
Draw and as well as point O, Q and P are collinear.
Clearly, PQ = 3 cm
Let then
In we have
…… 1
And
…… 2
From 1
and 2
we get
? 6x +
61
4
=
121
4
? 6x =
121-61
4
? 6x =
60
4
? x =
5
2
Putting the value of x in 2
we get,
Q u e s t i o n : 1 5
Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.
S o l u t i o n :
Let P is the mid point of chord AB of circle C(O, r) then according to question, line OQ passes through the point P.
Then prove that OQ bisect the arc AB.
Join OA and OB.
In ? AOP and ? BOP
Radiiofthesamecircle
(P is the mid point of chord AB)
Common
Therefore,
bycpct
Thus
Arc AQ = arc BQ
Therefore,
Hence Proved.
Q u e s t i o n : 1 6
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FAQs on Circles- 2 RD Sharma Solutions - Mathematics (Maths) Class 9
| 1. What is the formula to find the circumference of a circle? |  |
Ans. The formula to find the circumference of a circle is C = 2πr, where C is the circumference and r is the radius of the circle.
| 2. How do I find the area of a circle? | |
Ans. The formula to find the area of a circle is A = πr^2, where A is the area and r is the radius of the circle.
| 3. What is the difference between a diameter and a radius of a circle? | |
Ans. The diameter of a circle is a straight line segment that passes through the center of the circle and connects two points on the circle's circumference. The radius of a circle is a straight line segment that connects the center of the circle to any point on the circle's circumference. In simple terms, the diameter is twice the length of the radius.
| 4. How do I find the radius of a circle if I know the circumference? | |
Ans. To find the radius of a circle if you know the circumference, you can use the formula r = C/(2π), where r is the radius and C is the circumference.
| 5. How do I find the circumference of a circle if I know the diameter? | |
Ans. To find the circumference of a circle if you know the diameter, you can use the formula C = πd, where C is the circumference and d is the diameter of the circle.
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Nov 24, 2024 Last updated |
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