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

 Page 1


1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose
sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
Page 2


1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose
sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
 
Sol:
We know that when a quadrilateral circumscribes a circle then sum of opposites sides is 
equal to the sum of other opposite sides.
6 8 9
5
AB CD AD BC
AD
AD cm
2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB
then what is the measure of . OAB
Sol:
Construction: Join OB
We know that the radius and tangent are perpendicular at their point of contact
90 OBP OAP
Now, In quadrilateral AOBP
360 AOB OBP APB OAP [Angle sum property of a quadrilateral]
90 50 90 360
230 360
130
AOB
BOC
AOB
Now, In isosceles triangle AOB
180 AOB OAB OBA [Angle sum property of a triangle]
130 2 180
25
OAB OAB OBA
OAB
3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an
external point P. If 70 TPQ , find the . TRQ
Page 3


1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose
sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
 
Sol:
We know that when a quadrilateral circumscribes a circle then sum of opposites sides is 
equal to the sum of other opposite sides.
6 8 9
5
AB CD AD BC
AD
AD cm
2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB
then what is the measure of . OAB
Sol:
Construction: Join OB
We know that the radius and tangent are perpendicular at their point of contact
90 OBP OAP
Now, In quadrilateral AOBP
360 AOB OBP APB OAP [Angle sum property of a quadrilateral]
90 50 90 360
230 360
130
AOB
BOC
AOB
Now, In isosceles triangle AOB
180 AOB OAB OBA [Angle sum property of a triangle]
130 2 180
25
OAB OAB OBA
OAB
3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an
external point P. If 70 TPQ , find the . TRQ
 
Sol:
Construction: Join OQ and OT
We know that the radius and tangent are perpendicular at their point of contact
90 OTP OQP
Now, In quadrilateral OQPT
360 QOT OTP OQP TPO [Angle sum property of a quadrilateral]
90 90 70 360
250 360
110
QOT
QOT
QOT
We know that the angle subtended by an arc at the center is double the angle subtended by 
the arc at any point on the remaining part of the circle.
1
55
2
TRQ QOT
4. In the given figure common tangents AB and CD to the two circles with centres 
1 2
O and O
intersect at E. Prove that AB=CD.
Sol:
We know that tangent segments to a circle from the same external point are congruent.
So, we have
EA = EC for the circle having center
1
O
and
ED = EB for the circle having center 
1
O
Now, Adding ED on both sides in EA = EC. we get
EA ED EC ED
EA EB EC ED
AB CD
Page 4


1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose
sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
 
Sol:
We know that when a quadrilateral circumscribes a circle then sum of opposites sides is 
equal to the sum of other opposite sides.
6 8 9
5
AB CD AD BC
AD
AD cm
2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB
then what is the measure of . OAB
Sol:
Construction: Join OB
We know that the radius and tangent are perpendicular at their point of contact
90 OBP OAP
Now, In quadrilateral AOBP
360 AOB OBP APB OAP [Angle sum property of a quadrilateral]
90 50 90 360
230 360
130
AOB
BOC
AOB
Now, In isosceles triangle AOB
180 AOB OAB OBA [Angle sum property of a triangle]
130 2 180
25
OAB OAB OBA
OAB
3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an
external point P. If 70 TPQ , find the . TRQ
 
Sol:
Construction: Join OQ and OT
We know that the radius and tangent are perpendicular at their point of contact
90 OTP OQP
Now, In quadrilateral OQPT
360 QOT OTP OQP TPO [Angle sum property of a quadrilateral]
90 90 70 360
250 360
110
QOT
QOT
QOT
We know that the angle subtended by an arc at the center is double the angle subtended by 
the arc at any point on the remaining part of the circle.
1
55
2
TRQ QOT
4. In the given figure common tangents AB and CD to the two circles with centres 
1 2
O and O
intersect at E. Prove that AB=CD.
Sol:
We know that tangent segments to a circle from the same external point are congruent.
So, we have
EA = EC for the circle having center
1
O
and
ED = EB for the circle having center 
1
O
Now, Adding ED on both sides in EA = EC. we get
EA ED EC ED
EA EB EC ED
AB CD
 
5. If PT is a tangent to a circle with center O and PQ is a chord of the circle such that
70 QPT , then find the measure of . POQ
Sol:
We know that the radius and tangent are perpendicular at their point of contact.
90 OPT
Now, 90 70 20 OPQ OPT TPQ
Since, OP = OQ as both are radius
20 OPQ OQP (Angles opposite to equal sides are equal)
Now, In isosceles POQ
180 POQ OPQ OQP (Angle sum property of a triangle)
180 20 140 POQ
6. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such
that the segments BD and DC into which BC is divided by the point of contact D, are of
lengths 4cm and 3cm respectively. If the area of 
2
21 ABC cm then find the lengths of 
sides AB and AC.
Sol:
Construction: Join OA, OB, OC, OE AB at E and OF AC at F
We know that tangent segments to a circle from the same external point are congruent
Now, we have
, 4 AE AF BD BE cm and 3 CD CF cm
Now,
Area ABC Area BOC Area AOB Area AOC
Page 5


1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose
sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
 
Sol:
We know that when a quadrilateral circumscribes a circle then sum of opposites sides is 
equal to the sum of other opposite sides.
6 8 9
5
AB CD AD BC
AD
AD cm
2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB
then what is the measure of . OAB
Sol:
Construction: Join OB
We know that the radius and tangent are perpendicular at their point of contact
90 OBP OAP
Now, In quadrilateral AOBP
360 AOB OBP APB OAP [Angle sum property of a quadrilateral]
90 50 90 360
230 360
130
AOB
BOC
AOB
Now, In isosceles triangle AOB
180 AOB OAB OBA [Angle sum property of a triangle]
130 2 180
25
OAB OAB OBA
OAB
3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an
external point P. If 70 TPQ , find the . TRQ
 
Sol:
Construction: Join OQ and OT
We know that the radius and tangent are perpendicular at their point of contact
90 OTP OQP
Now, In quadrilateral OQPT
360 QOT OTP OQP TPO [Angle sum property of a quadrilateral]
90 90 70 360
250 360
110
QOT
QOT
QOT
We know that the angle subtended by an arc at the center is double the angle subtended by 
the arc at any point on the remaining part of the circle.
1
55
2
TRQ QOT
4. In the given figure common tangents AB and CD to the two circles with centres 
1 2
O and O
intersect at E. Prove that AB=CD.
Sol:
We know that tangent segments to a circle from the same external point are congruent.
So, we have
EA = EC for the circle having center
1
O
and
ED = EB for the circle having center 
1
O
Now, Adding ED on both sides in EA = EC. we get
EA ED EC ED
EA EB EC ED
AB CD
 
5. If PT is a tangent to a circle with center O and PQ is a chord of the circle such that
70 QPT , then find the measure of . POQ
Sol:
We know that the radius and tangent are perpendicular at their point of contact.
90 OPT
Now, 90 70 20 OPQ OPT TPQ
Since, OP = OQ as both are radius
20 OPQ OQP (Angles opposite to equal sides are equal)
Now, In isosceles POQ
180 POQ OPQ OQP (Angle sum property of a triangle)
180 20 140 POQ
6. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such
that the segments BD and DC into which BC is divided by the point of contact D, are of
lengths 4cm and 3cm respectively. If the area of 
2
21 ABC cm then find the lengths of 
sides AB and AC.
Sol:
Construction: Join OA, OB, OC, OE AB at E and OF AC at F
We know that tangent segments to a circle from the same external point are congruent
Now, we have
, 4 AE AF BD BE cm and 3 CD CF cm
Now,
Area ABC Area BOC Area AOB Area AOC
 
1 1 1
21
2 2 2
42 7 2 4 2 3 2
21 7 4 3
BC OD AB OE AC OF
x x
x x
21 14 2
2 7
3.5
4 3.5 7.5 3 3.5 6.5
x
x
x cm
AB cm and AC cm
7. Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger
circle  (in cm) which touches the smaller circle.
Sol:
Given Two circles have the same center O and AB is a chord of the larger circle touching 
the smaller circle at C; also. 5 OA cm and 3 OC cm
In 
2 2 2
, OAC OA OC AC
2 2 2
2 2 2
2
2
5 3
25 9
16
4
AC OA OC
AC
AC
AC
AC cm
2 AB AC (Since perpendicular drawn from the center of the circle bisects the chord)
2 4 8 AB cm
The length of the chord of the larger circle is 8 cm.
8. Prove that the perpendicular at the point of contact of the tangent to a circle passes through
the centre.
Sol: