RS Aggarwal Solutions Chapter 13 Construction Exercise 13B Class 10 Maths
Chapter Name  RS Aggarwal Chapter 13 Construction 
Book Name  RS Aggarwal Mathematics for Class 10 
Other Exercises 

Related Study  NCERT Solutions for Class 10 Maths 
Exercise 13B Solutions
1. Draw a circle of radius 3 cm. From 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.
Solution
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 OP =
7 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’.
PT = PT’ = 6.3 cm (Approx)
2. Draw two tangents to a circle of radius 3.5 cm from a
point P at a distance of 6.2 cm from its centre.
Solution
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 OP =
6.2 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’.
3. Draw a circle of radius 3.5 cm. Take two point 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.
Solution
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 OA = OB = 5 cm.
Step 3: Draw the perpendicular bisectors of OA and OB. Let
C and D be the midpoints of OA and OB, respectively.
Step 4: Draw a circle with C as center and radius OC (or
AC), to intersect the circle with centre 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.
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.
Solution
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 ∠OAX = 90°. Produce XA = Y.
Step
4: At B, draw ∠OBX’
= 90°. 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 from the point P to the circle.
Solution
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’S.
Solution
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 BP’. Also, join AQ and AQ’.
Here, AQ and AQ’ are the tangents from A to the circle with center B. Also, BP and BP’ 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°.
Solution
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 ∠BOC = 45°, Such that the radius OC meets the circle at C.
Step 4: 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°.
Solution
Steps of Construction:
Step 1: Draw a circle with center O and radius 3 cm.
Step 2: Draw any diameter AOB of the circle.
Step 3: Construct ∠BOC = 60° 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 angle of 60°.
9. Draw a circle of radius 32 cm. Draw a tangent to the circle making an angle of 30° with a line passing through the centre.
Solution
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 ∠AOP = 60°
Step 4: Draw PQ ⊥ OP, meeting OB at Q.
Step 5: Then, PQ is the desired tangent, such that ∠OQP = 30°.
10. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.
Solution
Steps of Construction:
Step 1: Mark a point O on the paper.
Step 2: With O as center and radii 4 cm and 6 cm, draw two concentric circles.
Step 3: Mark a point P on the outer circle.
Step 4: Join OP.
Step 5: Draw the perpendicular bisector XY of OP, cutting OP at Q.
Step 6: Draw a circle with Q as center and radius OQ (or PQ), to intersect the inner circle in points T and T’.
Step 7: Join PT and PT’.
Here, PT and PT’ are the required tangents.
PT = PT’ = 4.5 cm (Approx)
Verification by actual calculation
Join OT to form a right ∆OTP (Radius is perpendicular to the tangent at the point of contact)
In right ∆OTP,
(OP = 6 cm and OT = 4 cm)
11. Draw a line segment AB of lengths 5. 4 cm. Divide it into six equal parts, Write the steps of construction.
Solution
Steps of Construction:
Step 1: Draw a line segment AB = 5.4 cm.
Step 2: Draw a ray AX, making an acute angle, ∠BAX.
Step 3: Jong AX, mark 6 points, A_{1}, A_{2}, A_{3}, A_{4}, A_{5} such that,
AA_{1} = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4} = A_{4}A_{5} = A_{5}A_{6}.
Step 4: Join A_{6}B.
Step 5: Draw A_{1}C, A_{2}D, A_{3}D, A_{4}F and A_{5}A_{6}.
12. Draw a line segment AB of length 6.5 cm and divided it in the ratio 4 : 7. Measure each of the two parts.
Solution
Steps of Construction.
Step 1: Draw a line segment AB = 6.5 cm.
Step 2: Draw a ray AX, making an acute angle ∠BAX.
Step 3: Join AX, mark (4 + 7) = 11 points A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}, A_{7}, A_{8}, A_{9}, A_{10}, A_{11} such that AA_{1} = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4} = A_{3}A_{4 }= A_{4}A_{5} = A_{5}A_{6} = A_{6}A_{7}= A_{7}A_{8} = A_{8}A_{9} = A_{9}A_{10} = A_{10}A_{11}
Step 4: Join A_{11}B. .
Step 5: From A_{3}, draw A_{4}C  A_{11}B, meeting AB at C.
Thus, C is the point on AB, which divides it in the ratio 4 : 7.
From the figure,
AC = 2.36 cm
CB = 4.14 cm
13. Construct a ∆ABC in which B = 6.5 cm, AB = 4.5 cm and ∠ABC = 60°
Solution
Steps of Construction:
Step 1: Draw a line segment BC = 6.5 cm.
Step 2: With B as center, draw an angle of 60°.
Step 3: With B as center and radius equal to 4.5 cm, draw an arc, cutting the angle at A
Step 4: Join AB and AC.
Thus, ∆ABC is obtained.
Step 5: Below BC, draw an acute ∠CBX.
Step 6: Along BX, mark off four points B_{1}, B_{2}, B_{3}, B_{4} such that BB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4}
Step 7: Join B_{4}C.
Step 8: From B_{3}, draw B_{3}D  B_{4}C meeting BC at D.
Step 9: From D, draw DE  CA, meeting AB at E.
Solution
Steps of Construction:
Step 1: Draw the line l.
Step 2: Draw an angle of 90° at M on l
Step 3: Cut an arc of radius 3 cm on the perpendicular. Mark the point as A.
Step 4: With A as center, make an angle of 30° and let it cut l at C. We get ∠ACB = 60°.
Step 5: Cut an arc of 5 cm from C on l and mark the point as B.
Step 6: Join AB.
Thus, ∆ABC is obtained.
Step 7: Extend AB to D, such that BD = BC.
Step 8: Draw DE  BC, cutting AC produced to E.
15. Construct an isosceles triangle whose base is 9 cm and altitude 5 cm. Construct another triangle whose sides are 3/4 of the corresponding sides of the first isosceles triangle.
Solution
Steps of Construction:
Step 1: Draw a line segment BC = 9 cm
Step 2: With B as center, draw an arc each above and below BC.
Step 3: With C as center, draw an arc each above and below BC.
Step 4: Join their points of intersection to obtain the perpendicular bisector of BC. Let it intersect BC at D.
Step 5: From D, cut an arc of radius 5 cm and mark the points as A
Step 6: Join AB and AC
Thus ∆ABC is obtained.
Step 7: Below BC make an acute ∠CBX.
Step 8: Along BX, mark off four points B_{1}, B_{2}, B_{3}, B_{4} such that BB_{1} = B_{1}B_{2 }= B_{2}B_{3} = B3B4
Step 9: Join B_{4}C.
Step 10: From B_{3}, draw B_{2}E  B_{4}C meeting BC at E.
Step 11: From E, draw EF  CA meeting AB at F.
16. Draw a ∆ABC, rightangled at B such that AB = 3 cm and BC = 4 cm. Now, Construct a triangle similar to ∆ABC, each whose sides is 7/5 times the corresponding side of ∆ABC.
Solution
Steps of Construction
Step 1: Draw a line segment BC = 4 cm
Step 2: With B as center draw an angle of 90°
Step 3: With B as center and radius equal to 3 cm cut an arc at the right angle and name it A
Step 4: Join AB and AC.
Thus, ∆ABC is obtained.
Step 5: Extend BC to D, such that BD = 7/5 BC = 7/5(4) cm) = 5.6 cm
Step 6: Draw DE  CA cutting AB produced to E
Thus, ∆EBD is the required triangle, each of whose sides is 7/5 the corresponding sides of ∆ABC.
17. Draw a circle of radius 4.8 cm. Take a point P on it. Without using the centre of the circle, construct a tangent at the point P. Write the steps of Construction.
Solution
Steps of Construction:
Step 1: Draw a circle of radius 4.8 cm.
Step 2: Mark a point P on it:
Step 3: Draw any chord PQ.
Step 4: Take a point R on the major arc QP
Step 5: Join PR and RQ
Step 6: Draw ∠QPT = ∠PRQ
Step 7: Produce TP to T’ as shown in the figure.
18. Draw a circle of radius 3.5 cm. Draw a pair of tangents to this circle which are inclined to each other at an angle of 60°. Write the steps of construction.
Solution
Steps of Construction:
Step 1: Draw a circle with center O and radius = 3.5 cm.
Step 2: Draw any diameter AOB of this circle
Step 3: Construct ∠BOC = 60°, such that the radius OC meets the circle at C.
Step 4: Draw MA ⊥ AB and NC ⊥ OC.
Let AM and CN intersect at P.
Then, PA and PC are the required tangents to the given circle that are inclined at an angle of 60°.
19. Draw a circle of radius 4 cm. Draw tangent to the circle making an angle of 60° with a line passing through the centre.
Solution
Steps of Construction:
Step 1: Draw a circle with centre O and radius 4 cm
Step 2: Draw radius OA and produce it to B.
Step 3: Make ∠AOP = 30°
Step 4: Draw PQ ⊥ OP, meeting OB at Q.
Step 5: Then, PQ is the desired tangent, such that ∠OQP = 60°
Solution
Steps of Construction:
Step 1: Draw a circle with O as center and radius 6 cm
Step 2: Draw another circle with O as center and radius 4 cm.
Step 3: Mark a point P on the circle with radius 6 cm
Step 4: Join OP and bisect it at M.
Step 5: Draw a circle with M as center and radius equal to MP to intersect the given circle with radius 4 cm at points T and T’.
Step 6: Join PT and PT’.
Thus, PT or PT’ the required tangents and measure 4.4 cm each.