Circles (Exercise 10A) RD Sharma Solutions | Mathematics (Maths) Class 10 PDF Download

 Page 1


 
1. Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from 
the center of the circle
Sol:
Let O be the center of the given circle.
Let P be a point, such that
OP = 17 cm.
Let OT be the radius, where
OT = 5cm
Join TP, where TP is a tangent.
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2
2 2
17 8
289 64
225
15
TP OP OT
cm
The length of the tangent is 15 cm.
2. A point P is 25 cm away from the center of a circle and the length of tangent drawn from P 
to the circle is 24 cm. Find the radius of the circle.
Sol:
Draw a circle and let P be a point such that OP = 25cm.
Let TP be the tangent, so that TP = 24cm
Join OT where OT is radius.
Page 2


 
1. Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from 
the center of the circle
Sol:
Let O be the center of the given circle.
Let P be a point, such that
OP = 17 cm.
Let OT be the radius, where
OT = 5cm
Join TP, where TP is a tangent.
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2
2 2
17 8
289 64
225
15
TP OP OT
cm
The length of the tangent is 15 cm.
2. A point P is 25 cm away from the center of a circle and the length of tangent drawn from P 
to the circle is 24 cm. Find the radius of the circle.
Sol:
Draw a circle and let P be a point such that OP = 25cm.
Let TP be the tangent, so that TP = 24cm
Join OT where OT is radius.
 
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2 2
2 2
25 24
625 576
49
7
OT OP TP
cm
The length of the radius is 7cm.
3. Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the 
larger circle which touches the smaller circle.
Sol:
We know that the radius and tangent are perpendicular at their point of contact 
In right triangle AOP
2 2 2
2 2
2
2
6.5 2.5
36
6
AO OP PA
PA
PA
PA cm
Since, the perpendicular drawn from the center bisects the chord.
6 PA PB cm
Now, 6 6 12 AB AP PB cm
Hence, the length of the chord of the larger circle is 12cm.
4. In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC 
at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length 
of AD, BE and CF.
Page 3


 
1. Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from 
the center of the circle
Sol:
Let O be the center of the given circle.
Let P be a point, such that
OP = 17 cm.
Let OT be the radius, where
OT = 5cm
Join TP, where TP is a tangent.
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2
2 2
17 8
289 64
225
15
TP OP OT
cm
The length of the tangent is 15 cm.
2. A point P is 25 cm away from the center of a circle and the length of tangent drawn from P 
to the circle is 24 cm. Find the radius of the circle.
Sol:
Draw a circle and let P be a point such that OP = 25cm.
Let TP be the tangent, so that TP = 24cm
Join OT where OT is radius.
 
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2 2
2 2
25 24
625 576
49
7
OT OP TP
cm
The length of the radius is 7cm.
3. Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the 
larger circle which touches the smaller circle.
Sol:
We know that the radius and tangent are perpendicular at their point of contact 
In right triangle AOP
2 2 2
2 2
2
2
6.5 2.5
36
6
AO OP PA
PA
PA
PA cm
Since, the perpendicular drawn from the center bisects the chord.
6 PA PB cm
Now, 6 6 12 AB AP PB cm
Hence, the length of the chord of the larger circle is 12cm.
4. In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC 
at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length 
of AD, BE and CF.
 
Sol:
We know that tangent segments to a circle from the same external point are congruent.
Now, we have
AD = AF, BD = BE and CE = CF
Now, AD + BD = l2cm
AF + FC = l0 cm
AD + FC = l0 cm
BE + EC = 8 cm
BD + FC = 8cm (3)
Adding all these we get
AD + BD + AD + FC + BD + FC = 30
2(AD + BD + FC) = 30
AD + BD + FC = l5cm (4)
Solving (1) and (4), we get
FC = 3 cm
Solving (2) and (4), we get
BD = 5 cm
Solving (3) and (4), we get
and AD = 7 cm
AD = AF =7 cm, BD = BE = 5 cm and CE = CF =3 cm
5. In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three 
sides are AB = 6cm, BC=7cm and CD=4 cm. Find AD.
Sol:
Let the circle touch the sides of the quadrilateral AB, BC, CD and DA at P, Q, R and S
respectively.
Page 4


 
1. Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from 
the center of the circle
Sol:
Let O be the center of the given circle.
Let P be a point, such that
OP = 17 cm.
Let OT be the radius, where
OT = 5cm
Join TP, where TP is a tangent.
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2
2 2
17 8
289 64
225
15
TP OP OT
cm
The length of the tangent is 15 cm.
2. A point P is 25 cm away from the center of a circle and the length of tangent drawn from P 
to the circle is 24 cm. Find the radius of the circle.
Sol:
Draw a circle and let P be a point such that OP = 25cm.
Let TP be the tangent, so that TP = 24cm
Join OT where OT is radius.
 
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2 2
2 2
25 24
625 576
49
7
OT OP TP
cm
The length of the radius is 7cm.
3. Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the 
larger circle which touches the smaller circle.
Sol:
We know that the radius and tangent are perpendicular at their point of contact 
In right triangle AOP
2 2 2
2 2
2
2
6.5 2.5
36
6
AO OP PA
PA
PA
PA cm
Since, the perpendicular drawn from the center bisects the chord.
6 PA PB cm
Now, 6 6 12 AB AP PB cm
Hence, the length of the chord of the larger circle is 12cm.
4. In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC 
at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length 
of AD, BE and CF.
 
Sol:
We know that tangent segments to a circle from the same external point are congruent.
Now, we have
AD = AF, BD = BE and CE = CF
Now, AD + BD = l2cm
AF + FC = l0 cm
AD + FC = l0 cm
BE + EC = 8 cm
BD + FC = 8cm (3)
Adding all these we get
AD + BD + AD + FC + BD + FC = 30
2(AD + BD + FC) = 30
AD + BD + FC = l5cm (4)
Solving (1) and (4), we get
FC = 3 cm
Solving (2) and (4), we get
BD = 5 cm
Solving (3) and (4), we get
and AD = 7 cm
AD = AF =7 cm, BD = BE = 5 cm and CE = CF =3 cm
5. In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three 
sides are AB = 6cm, BC=7cm and CD=4 cm. Find AD.
Sol:
Let the circle touch the sides of the quadrilateral AB, BC, CD and DA at P, Q, R and S
respectively.
 
Given, AB = 6cm, BC = 7 cm and CD = 4cm.
Tangents drawn from an external point are equal.
AP = AS, BP = BQ,CR = CQ and DR = DS
Now, AB + CD (AP + BP) + (CR + DR)
6 4 7
3 .
AB CD AS BQ CQ DS
AB CD AS DS BQ CQ
AB CD AD BC
AD AB CD BC
AD
AD cm
The length of AD is 3 cm.
6. In the given figure, the chord AB of the larger of the two concentric circles, with center O,
touches the smaller circle at C. Prove that AC = CB.
Sol:
Construction: Join OA, OC and OB
We know that the radius and tangent are perpendicular at their point of contact
90 OCA OCB
Now, In OCA and OCB
90 OCA OCB
OA OB (Radii of the larger circle)
OC OC (Common)
By RHS congruency
OCA OCB
CA CB
Page 5


 
1. Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from 
the center of the circle
Sol:
Let O be the center of the given circle.
Let P be a point, such that
OP = 17 cm.
Let OT be the radius, where
OT = 5cm
Join TP, where TP is a tangent.
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2
2 2
17 8
289 64
225
15
TP OP OT
cm
The length of the tangent is 15 cm.
2. A point P is 25 cm away from the center of a circle and the length of tangent drawn from P 
to the circle is 24 cm. Find the radius of the circle.
Sol:
Draw a circle and let P be a point such that OP = 25cm.
Let TP be the tangent, so that TP = 24cm
Join OT where OT is radius.
 
Now, tangent drawn from an external point is perpendicular to the radius at the point of 
contact.
OT PT
In the right , OTP we have:
2 2 2
OP OT TP
2 2 2
2 2
25 24
625 576
49
7
OT OP TP
cm
The length of the radius is 7cm.
3. Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the 
larger circle which touches the smaller circle.
Sol:
We know that the radius and tangent are perpendicular at their point of contact 
In right triangle AOP
2 2 2
2 2
2
2
6.5 2.5
36
6
AO OP PA
PA
PA
PA cm
Since, the perpendicular drawn from the center bisects the chord.
6 PA PB cm
Now, 6 6 12 AB AP PB cm
Hence, the length of the chord of the larger circle is 12cm.
4. In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC 
at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length 
of AD, BE and CF.
 
Sol:
We know that tangent segments to a circle from the same external point are congruent.
Now, we have
AD = AF, BD = BE and CE = CF
Now, AD + BD = l2cm
AF + FC = l0 cm
AD + FC = l0 cm
BE + EC = 8 cm
BD + FC = 8cm (3)
Adding all these we get
AD + BD + AD + FC + BD + FC = 30
2(AD + BD + FC) = 30
AD + BD + FC = l5cm (4)
Solving (1) and (4), we get
FC = 3 cm
Solving (2) and (4), we get
BD = 5 cm
Solving (3) and (4), we get
and AD = 7 cm
AD = AF =7 cm, BD = BE = 5 cm and CE = CF =3 cm
5. In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three 
sides are AB = 6cm, BC=7cm and CD=4 cm. Find AD.
Sol:
Let the circle touch the sides of the quadrilateral AB, BC, CD and DA at P, Q, R and S
respectively.
 
Given, AB = 6cm, BC = 7 cm and CD = 4cm.
Tangents drawn from an external point are equal.
AP = AS, BP = BQ,CR = CQ and DR = DS
Now, AB + CD (AP + BP) + (CR + DR)
6 4 7
3 .
AB CD AS BQ CQ DS
AB CD AS DS BQ CQ
AB CD AD BC
AD AB CD BC
AD
AD cm
The length of AD is 3 cm.
6. In the given figure, the chord AB of the larger of the two concentric circles, with center O,
touches the smaller circle at C. Prove that AC = CB.
Sol:
Construction: Join OA, OC and OB
We know that the radius and tangent are perpendicular at their point of contact
90 OCA OCB
Now, In OCA and OCB
90 OCA OCB
OA OB (Radii of the larger circle)
OC OC (Common)
By RHS congruency
OCA OCB
CA CB
 
7. From an external point P, tangents PA and PB are drawn to a circle with center O. If CD is 
the tangent to the circle at a  point E and PA = 14cm, find the  perimeter of PCD .
Sol:
Given, PA and PB are the tangents to a circle with center O and CD is a tangent at E and 
PA = 14 cm.
Tangents drawn from an external point are equal.
PA = PB, CA = CE and DB = DE
Perimeter of PCD PC CD PD
2
2 14
28
PA CA CE DE PB DB
PA CE CE DE PB DE
PA PB
PA PA PB
cm
cm
=28 cm
Perimeter of 28 . PCD cm
8. A circle is inscribed in a ABC touching AB, BC and AC at P, Q and R respectively. If 
AB = 10 cm, AR=7cm and CR=5cm, find the length of BC.
Sol:
Given, a circle inscribed in triangle ABC, such that the circle touches the sides of the 
triangle
Tangents drawn to a circle from an external point are equal.
7 , 5 . AP AR cm CQ CR cm
Now, 10 7 3 BP AB AP cm
3 BP BQ cm 
BC BQ QC