# Class 9 Maths Chapter 11

### Constructions

Exercise 11.1
Question 1.
Construct an angle of 90° at the initial point of a given ray and justify the construction.

Solution:
Steprf of Construction:
Step I : Draw .
Step II : Taking O as centre and having a suitable radius, draw a semicircle, which cuts at B.
Step III : Keeping the radius same, divide the semicircle into three equal parts such that .
Step IV : Draw and .
Step V : Draw , the bisector of ∠COD. Thus, ∠AOF = 90°
Justification:
∵ O is the centre of the semicircle and it is divided into 3 equal parts.

⇒ ∠BOC = ∠COD = ∠DOE [Equal chords subtend equal angles at the centre]
And, ∠BOC + ∠COD + ∠DOE = 180°
⇒ ∠BOC + ∠BOC + ∠BOC = 180°
⇒ 3∠BOC = 180°
⇒ ∠BOC = 60°
Similarly, ∠COD = 60° and ∠DOE = 60°
is the bisector of ∠COD
∴ ∠COF = ∠COD = (60°) = 30°
Now, ∠BOC + ∠COF = 60° + 30°
⇒ ∠BOF = 90° or ∠AOF = 90°

Question 2.
Construct an angle of 45° at the initial point of a given ray and justify the construction.

Solution:
Steps of Construction:
Stept I : Draw .
Step II : Taking O as centre and with a suitable radius, draw a semicircle such that it intersects . at B.
Step III : Taking B as centre and keeping the same radius, cut the semicircle at C. Now, taking C as centre and keeping the same radius, cut the semicircle at D and similarly, cut at E, such that
Step IV : Draw and .
Step V : Draw , the angle bisector of ∠BOC.
Step VI : Draw , the ajngle bisector of ∠FOC. Thus, ∠BOG = 45° or ∠AOG = 45°
Justification:

∴ ∠BOC = ∠COD = ∠DOE [Equal chords subtend equal angles at the centre]
Since, ∠BOC + ∠COD + ∠DOE = 180°
⇒ ∠BOC = 60°
is the bisector of ∠BOC.
∴ ∠COF = ∠BOC = (60°) = 30° …(1)
Also, is the bisector of ∠COF.
∠FOG = ∠COF = (30°) = 15° …(2)
Adding (1) and (2), we get
∠COF + ∠FOG = 30° + 15° = 45°
⇒ ∠BOF + ∠FOG = 45° [∵ ∠COF = ∠BOF]
⇒ ∠BOG = 45°

Question 3.
Construct the angles of the following measurements
(i) 30°
(ii) 22
(iii) 15°

Solution:
(i) Angle of 30°
Steps of Construction:
Step I : Draw .
Step II : With O as centre and having a suitable radius, draw an arc cutting at B.
Step III : With centre at B and keeping the same radius as above, draw an arc to cut the previous arc at C.
Step IV : Join which gives ∠BOC = 60°.
Step V : Draw , bisector of ∠BOC, such that ∠BOD = ∠BOC = (60°) = 30° Thus, ∠BOD = 30° or ∠AOD = 30°

(ii) Angle of 22
Steps of Construction:
Step I : Draw .
Step II : Construct ∠AOB = 90°
Step III : Draw , the bisector of ∠AOB, such that
∠AOC = ∠AOB = (90°) = 45°
Step IV : Now, draw OD, the bisector of ∠AOC, such that
∠AOD = ∠AOC = (45°) = 22 Thus, ∠AOD = 22

(iii) Angle of 15°
Steps of Construction:
Step I : Draw .
Step II : Construct ∠AOB = 60°.
Step III : Draw OC, the bisector of ∠AOB, such that
∠AOC = ∠AOB = (60°) = 30°
i.e., ∠AOC = 30°
Step IV : Draw OD, the bisector of ∠AOC such that
∠AOD = ∠AOC = (30°) = 15° Thus, ∠AOD = 15°

Question 4.
Construct the following angles and verify by measuring them by a protractor
(i) 75°
(ii) 105°
(iii) 135°

Solution:
Step I : Draw .
Step II : With O as centre and having a suitable radius, draw an arc which cuts at B.
Step III : With centre B and keeping the same radius, mark a point C on the previous arc.
Step IV : With centre C and having the same radius, mark another point D on the arc of step II.
Step V : Join and , which gives ∠COD = 60° = ∠BOC.
Step VI : Draw , the bisector of ∠COD, such that
∠COP = ∠COD = (60°) = 30°.
Step VII: Draw , the bisector of ∠COP, such that
∠COQ = ∠COP = (30°) = 15°. Thus, ∠BOQ = 60° + 15° = 75°∠AOQ = 75°

(ii) Steps of Construction:
Step I : Draw .
Step II : With centre O and having a suitable radius, draw an arc which cuts at B.
Step III : With centre B and keeping the same radius, mark a point C on the previous arc.
Step IV : With centre C and having the same radius, mark another point D on the arc drawn in step II.
Step V : Draw OP, the bisector of CD which cuts CD at E such that ∠BOP = 90°.
Step VI : Draw , the bisector of such that ∠POQ = 15° Thus, ∠AOQ = 90° + 15° = 105°

(iii) Steps of Construction:
Step I : Draw .
Step II : With centre O and having a suitable radius, draw an arc which cuts at A
Step III : Keeping the same radius and starting from A, mark points Q, R and S on the arc of step II such that .
StepIV :Draw , thebisector of which cuts the arc at T.
Step V : Draw , the bisector of . Thus, ∠POQ = 135°

Question 5.
Construct an equilateral triangle, given its side and justify the construction.

Solution:
pt us construct an equilateral triangle, each of whose side = 3 cm(say).
Steps of Construction:
Step I : Draw .
Step II : Taking O as centre and radius equal to 3 cm, draw an arc to cut at B such that OB = 3 cm
Step III : Taking B as centre and radius equal to OB, draw an arc to intersect the previous arc at C.
Step IV : Join OC and BC. Thus, ∆OBC is the required equilateral triangle.

Justification:
∵ The arcs and are drawn with the same radius.
=
⇒ OC = BC [Chords corresponding to equal arcs are equal]
∵ OC = OB = BC
∴ OBC is an equilateral triangle.

Exercise 11.2

Question 1.
Construct a ∆ ABC in which BC = 7 cm, ∠B = 75° and AB + AC = 13 cm.

Solution:
Steps of Construction:
Step I : Draw .
Step II : Along , cut off a line segment BC = 7 cm.
Step III : At B, construct ∠CBY = 75°
Step IV : From , cut off BD = 13 cm (= AB + AC)
Step V : Join DC.
Step VI : Draw a perpendicular bisector of CD which meets BD at A.
Step VII: Join AC. Thus, ∆ABC is the required triangle.

Question 2.
Construct a ABC in which BC = 8 cm, ∠B = 45° and AB – AC = 35 cm.

Solution:
Steps of Construction:
Step I : Draw .
Step II : Along , cut off a line segment BC = 8 cm.
Step III : At B, construct ∠CBY = 45°
Step IV : From , cut off BD = 3.5 cm (= AB – AC)
Step V : Join DC.
Step VI : Draw PQ, perpendicular bisector of DC, which intersects at A.
Step VII: Join AC. Thus, ∆ABC is the required triangle.

Question 3.
Construct a ∆ ABC in which QR = 6 cm, ∠Q = 60° and PR – PQ = 2 cm.

Solution:
Steps of Construction:
Step I : Draw .
Step II : Along , cut off a line segment QR = 6 cm.
Step III : Construct a line YQY’ such that ∠RQY = 60°.
Step IV : Cut off QS = 2 cm (= PR – PQ) on QY’.
Step V : Join SR.
Step VI : Draw MN, perpendicular bisector of SR, which intersects QY at P.
Step VII: Join PR. Thus, ∆PQR is the required triangle.

Question 4.
Construct a ∆ XYZ in which ∠Y = 30°, ∠Y = 90° and XY + YZ + ZX = 11 cm.

Solution:
Steps of Construction:
Step I : Draw a line segment AB = 11 cm = (XY+YZ + ZX)
Step II : Construct ∠BAP = 30°
Step III : Construct ∠ABQ = 90°
Step IV : Draw AR, the bisector of ∠BAP.
Step V : Draw BS, the bisector of ∠ABQ. Let AR and BS intersect at X.
Step VI : Draw perpendicular bisector of , which intersects AB at Y.
Step VII: Draw perpendicular bisector of , which intersects AB at Z.
Step VIII: Join XY and XZ. Thus, ∆XYZ is the required triangle.

Question 5.
Construct a right triangle whose base is 12 cm and sum of its hypotenuse and other side is 18 cm.

Solution:
Steps of Construction:
Step I : Draw BC = 12 cm.
Step II : At B, construct ∠CBY = 90°.
Step III : Along , cut off a line segment BX = 18 cm.
Step IV : Join CX.
Step V : Draw PQ, perpendicular bisector of CX, which meets BX at A.
Step VI : Join AC. Thus, ∆ABC is the required triangle.