Constructions (Exercise 11B) RD Sharma Solutions | Mathematics (Maths) Class 10 PDF Download

 Page 1


1. Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw
two tangents to the circle. Also, measure the lengths of the tangents.
Sol: 
Steps of Construction 
Step 1: Draw a circle with O as center and radius 3 cm. 
Step 2: Mark a point P outside the circle such that 7 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP. cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or OQ), to intersect the given circle 
at the points T and T
Step 5: Join PT and PT . 
Here, PT and  are the required tangents. 
' 6.3 PT PT cm (Approx) 
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form
its centre.
Sol:
Steps of Construction
Step 1: Draw a circle with O as center and radius 3.5 cm.
Page 2


1. Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw
two tangents to the circle. Also, measure the lengths of the tangents.
Sol: 
Steps of Construction 
Step 1: Draw a circle with O as center and radius 3 cm. 
Step 2: Mark a point P outside the circle such that 7 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP. cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or OQ), to intersect the given circle 
at the points T and T
Step 5: Join PT and PT . 
Here, PT and  are the required tangents. 
' 6.3 PT PT cm (Approx) 
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form
its centre.
Sol:
Steps of Construction
Step 1: Draw a circle with O as center and radius 3.5 cm.
 
Step 2: Mark a point P outside the circle such that 6.2 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or 0Q), to intersect the given circle at 
the
Step 5: Join PT
3. Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter,
each at a distance of 5 cm from its center. Draw tangents to the circle from each of these
points A and B.
Sol:
Steps of Construction
Step 1: Draw a circle with center O and radius 3.5 cm.
Step 2: Extends its diameter on both sides and mark two points A and B on it such that
5 . OA OB cm
Step 3: Draw the perpendicular bisectors of OA and OB. Let C and D be the mid-points of
OA and OB, respectively.
Step 4: Draw a circle with C as center and radius OC (or AC), to intersect the circle with
center O, at the points P and Q.
Step 5: Draw another circle with D as center and radius OD (or BD), to intersect the circle
with center O at the points R and S.
Step 6: Join AP and AQ, Also, join BR and BS.
Here, AP and AQ are the tangents to the circle from A, Also, BR and BS are the tangents to 
the circle from B.
Page 3


1. Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw
two tangents to the circle. Also, measure the lengths of the tangents.
Sol: 
Steps of Construction 
Step 1: Draw a circle with O as center and radius 3 cm. 
Step 2: Mark a point P outside the circle such that 7 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP. cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or OQ), to intersect the given circle 
at the points T and T
Step 5: Join PT and PT . 
Here, PT and  are the required tangents. 
' 6.3 PT PT cm (Approx) 
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form
its centre.
Sol:
Steps of Construction
Step 1: Draw a circle with O as center and radius 3.5 cm.
 
Step 2: Mark a point P outside the circle such that 6.2 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or 0Q), to intersect the given circle at 
the
Step 5: Join PT
3. Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter,
each at a distance of 5 cm from its center. Draw tangents to the circle from each of these
points A and B.
Sol:
Steps of Construction
Step 1: Draw a circle with center O and radius 3.5 cm.
Step 2: Extends its diameter on both sides and mark two points A and B on it such that
5 . OA OB cm
Step 3: Draw the perpendicular bisectors of OA and OB. Let C and D be the mid-points of
OA and OB, respectively.
Step 4: Draw a circle with C as center and radius OC (or AC), to intersect the circle with
center O, at the points P and Q.
Step 5: Draw another circle with D as center and radius OD (or BD), to intersect the circle
with center O at the points R and S.
Step 6: Join AP and AQ, Also, join BR and BS.
Here, AP and AQ are the tangents to the circle from A, Also, BR and BS are the tangents to 
the circle from B.
 
4. Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle.
Construct tangents to the circle at each of the two end points of the diameter AB.
Sol:
Step 1: Draw a circle with center O and radius 4 cm.
Step 2: Draw any diameter AOB of the circle.
Step 3: At A, draw 90 . OAX Produce . XA Y
Step 4: At B, draw 90 . ' OBX Produce ' X B to '. Y
Here, ' ' XAY and X BY are the tangents to the circle at the end points of the diameter AB.
5. Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the
pair of tangents form the point P to the circle
Sol:
Steps of Construction
Step 1: Draw a circle with the help of a bangle.
Step 2: Mark a point P outside the circle.
Step 3: Through P. draw a secant PAB to intersect the circle at A and B.
Step 4: Produce AP to C such that PA = PC.
Step 5: Draw a semicircle with CB as diameter.
Step 6: Draw , PD BC intersecting the semicircle at D.
Step 7: With P as center and PD as radius, draw arcs to intersect the circle at T and T .
Step 8: Join PT and PT .
Here, PT and PT .
Page 4


1. Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw
two tangents to the circle. Also, measure the lengths of the tangents.
Sol: 
Steps of Construction 
Step 1: Draw a circle with O as center and radius 3 cm. 
Step 2: Mark a point P outside the circle such that 7 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP. cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or OQ), to intersect the given circle 
at the points T and T
Step 5: Join PT and PT . 
Here, PT and  are the required tangents. 
' 6.3 PT PT cm (Approx) 
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form
its centre.
Sol:
Steps of Construction
Step 1: Draw a circle with O as center and radius 3.5 cm.
 
Step 2: Mark a point P outside the circle such that 6.2 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or 0Q), to intersect the given circle at 
the
Step 5: Join PT
3. Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter,
each at a distance of 5 cm from its center. Draw tangents to the circle from each of these
points A and B.
Sol:
Steps of Construction
Step 1: Draw a circle with center O and radius 3.5 cm.
Step 2: Extends its diameter on both sides and mark two points A and B on it such that
5 . OA OB cm
Step 3: Draw the perpendicular bisectors of OA and OB. Let C and D be the mid-points of
OA and OB, respectively.
Step 4: Draw a circle with C as center and radius OC (or AC), to intersect the circle with
center O, at the points P and Q.
Step 5: Draw another circle with D as center and radius OD (or BD), to intersect the circle
with center O at the points R and S.
Step 6: Join AP and AQ, Also, join BR and BS.
Here, AP and AQ are the tangents to the circle from A, Also, BR and BS are the tangents to 
the circle from B.
 
4. Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle.
Construct tangents to the circle at each of the two end points of the diameter AB.
Sol:
Step 1: Draw a circle with center O and radius 4 cm.
Step 2: Draw any diameter AOB of the circle.
Step 3: At A, draw 90 . OAX Produce . XA Y
Step 4: At B, draw 90 . ' OBX Produce ' X B to '. Y
Here, ' ' XAY and X BY are the tangents to the circle at the end points of the diameter AB.
5. Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the
pair of tangents form the point P to the circle
Sol:
Steps of Construction
Step 1: Draw a circle with the help of a bangle.
Step 2: Mark a point P outside the circle.
Step 3: Through P. draw a secant PAB to intersect the circle at A and B.
Step 4: Produce AP to C such that PA = PC.
Step 5: Draw a semicircle with CB as diameter.
Step 6: Draw , PD BC intersecting the semicircle at D.
Step 7: With P as center and PD as radius, draw arcs to intersect the circle at T and T .
Step 8: Join PT and PT .
Here, PT and PT .
 
6. Draw a line segment AB of length 8 cm. Taking A as centre , draw a circle of radius 4 cm
and taking B as centre , draw another circle of radius 3 cm. Construct tangents to each
circle form the centre of the other circle.
Sol:
Steps of Construction
Step 1: Draw a line segment AB = 8 cm.
Step 2: With A as center and radius 4 cm, draw a circle.
Step 3: With B as center and radius 3 cm, draw another circle.
Step 4: Draw the perpendicular bisector XY of AB, cuffing AB at C.
Step 5: With C as center and radius AC (or BC), draw a circle intersecting the circle with
center A at P and P : and the circle with center B at Q and Q .
Step 6: Join BP and B Also, join AQ and AQ .
are the tangents from B to the circle with center A.
7. Draw a circle of radius 4.2. Draw a pair of tangents to this circle inclined to each other at
an angle of 45
Sol:
Steps of Construction:
Step 1: Draw a circle with center O and radius = 4.2 cm.
Step 2: Draw any diameter AOB of this circle.
Step 3: Construct 45 . BOC such that the radius OC meets the circle at C.
Step4: Draw AM AB and . CN OC
AM and CN intersect at P.
Thus, PA and PC are the required tangents to the given circle inclined at an angle of 45 .
Page 5


1. Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw
two tangents to the circle. Also, measure the lengths of the tangents.
Sol: 
Steps of Construction 
Step 1: Draw a circle with O as center and radius 3 cm. 
Step 2: Mark a point P outside the circle such that 7 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP. cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or OQ), to intersect the given circle 
at the points T and T
Step 5: Join PT and PT . 
Here, PT and  are the required tangents. 
' 6.3 PT PT cm (Approx) 
2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form
its centre.
Sol:
Steps of Construction
Step 1: Draw a circle with O as center and radius 3.5 cm.
 
Step 2: Mark a point P outside the circle such that 6.2 . OP cm
Step 3: Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.
Step 4: Draw a circle with Q as center and radius PQ (or 0Q), to intersect the given circle at 
the
Step 5: Join PT
3. Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter,
each at a distance of 5 cm from its center. Draw tangents to the circle from each of these
points A and B.
Sol:
Steps of Construction
Step 1: Draw a circle with center O and radius 3.5 cm.
Step 2: Extends its diameter on both sides and mark two points A and B on it such that
5 . OA OB cm
Step 3: Draw the perpendicular bisectors of OA and OB. Let C and D be the mid-points of
OA and OB, respectively.
Step 4: Draw a circle with C as center and radius OC (or AC), to intersect the circle with
center O, at the points P and Q.
Step 5: Draw another circle with D as center and radius OD (or BD), to intersect the circle
with center O at the points R and S.
Step 6: Join AP and AQ, Also, join BR and BS.
Here, AP and AQ are the tangents to the circle from A, Also, BR and BS are the tangents to 
the circle from B.
 
4. Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle.
Construct tangents to the circle at each of the two end points of the diameter AB.
Sol:
Step 1: Draw a circle with center O and radius 4 cm.
Step 2: Draw any diameter AOB of the circle.
Step 3: At A, draw 90 . OAX Produce . XA Y
Step 4: At B, draw 90 . ' OBX Produce ' X B to '. Y
Here, ' ' XAY and X BY are the tangents to the circle at the end points of the diameter AB.
5. Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the
pair of tangents form the point P to the circle
Sol:
Steps of Construction
Step 1: Draw a circle with the help of a bangle.
Step 2: Mark a point P outside the circle.
Step 3: Through P. draw a secant PAB to intersect the circle at A and B.
Step 4: Produce AP to C such that PA = PC.
Step 5: Draw a semicircle with CB as diameter.
Step 6: Draw , PD BC intersecting the semicircle at D.
Step 7: With P as center and PD as radius, draw arcs to intersect the circle at T and T .
Step 8: Join PT and PT .
Here, PT and PT .
 
6. Draw a line segment AB of length 8 cm. Taking A as centre , draw a circle of radius 4 cm
and taking B as centre , draw another circle of radius 3 cm. Construct tangents to each
circle form the centre of the other circle.
Sol:
Steps of Construction
Step 1: Draw a line segment AB = 8 cm.
Step 2: With A as center and radius 4 cm, draw a circle.
Step 3: With B as center and radius 3 cm, draw another circle.
Step 4: Draw the perpendicular bisector XY of AB, cuffing AB at C.
Step 5: With C as center and radius AC (or BC), draw a circle intersecting the circle with
center A at P and P : and the circle with center B at Q and Q .
Step 6: Join BP and B Also, join AQ and AQ .
are the tangents from B to the circle with center A.
7. Draw a circle of radius 4.2. Draw a pair of tangents to this circle inclined to each other at
an angle of 45
Sol:
Steps of Construction:
Step 1: Draw a circle with center O and radius = 4.2 cm.
Step 2: Draw any diameter AOB of this circle.
Step 3: Construct 45 . BOC such that the radius OC meets the circle at C.
Step4: Draw AM AB and . CN OC
AM and CN intersect at P.
Thus, PA and PC are the required tangents to the given circle inclined at an angle of 45 .
 
8. Write the steps of construction for drawing a pair of tangents to a circle of radius 3 cm ,
which are inclined to each other at an angle of  60 .
Sol:
Steps of Construction
Step 1: Draw a circle with center O and radius 3c m.
Step 2: Draw any diameter AOB of the circle.
Step 3: Construct 60 BOC such that radius OC cuts the circle at C.
Step 4: Draw AM AB and . CN OC Suppose AM and CN intersect each other at P.
Here, AP and CP are the pair of tangents to the circle inclined to each other at an angle of 
60°.
9. Draw a circle of radius 32 cm. Draw a tangent to the circle making an angle 30 with a line
passing through  the centre.
Sol:
Steps Of construction:
Step 1: Draw a circle with center O and radius 3 cm.
Step 2: Draw radius OA and produce it to B.
Step 3: Make 60 AOP
Step 4: Draw , PQ OP meeting OB at Q.
Step 5: Then, PQ is the desired tangent, such that 30 OQP
10. Construct a tangent to a circle of radius 4 cm form a point on the concentric circle of radius
6 cm and measure its length. Also, verify the measurement by actual calculation.
Sol:
Steps of Construction
Step 1: Mark a point O on the paper
Step 2: With O as center and radii 4cm and 6cm, draw two concentric circles.