### 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°

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.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°

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°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°

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°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°

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.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.

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.

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.

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.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.

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.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.

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.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.

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.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.

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.