Chapter 11 Constructions

Chapter 11 Constructions

 

Ex 11.1

Ex 11.1 Class 9 Maths 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°

Ex 11.1 Class 9 Maths Question 2.
Construct an angle of 45° at the initial point of a given ray and justify the construction.
Solution:
Steps of Construction:
Step 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°

Ex 11.1 Class 9 Maths 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°

Ex 11.1 Class 9 Maths 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 .
Step IV: Draw , the bisector of which cuts the arc at T.
Step V : Draw , the bisector of .

Thus, ∠POQ = 135°

Ex 11.1 Class 9 Maths 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.