Chapter 11: Constructions​ Exercise – 11.1

Question: 1

Determine a point which divides a line segment of length 12 cm internally in the ratio of 2: 3. Also, justify your construction. 

Solution:

Steps of Construction:

1. Draw a line segment AB of 12 cm

2. Through the points A and B draw two parallel line on the opposite side of AB

3. Cut 2 equal parts on AX and 3 equal parts on BY such that AX₁ = X₁X₂ and BX₁ = Y₁Y₂ = Y₂Y₃.

4. Join X₂Y₃ which intersects AB at P

∴  AP/PB = 2/3.

Justification:

In ΔAX₂P and ΔBY₃P, we have

∠APX₂ = ∠BPY₃ {Because they are vertically opposite angle}

∠X₂AP = ∠Y₃BP {Because they are alternate interior angles}

ΔAX₂P ΔBY₃P {Because AA similarity}

∴  AP/BP = AX₂/BY₃ = 2/3 {Because of C.P.C.T}

 

​Question: 2

Divide a line segment of length 9 cm internally in the ratio 4 : 3. Also, give justification for the construction.

Solution:

Steps of construction:

1. Draw a line segment AB of 9 cm

2. Through the points, A and B, draw two parallel lines AX and BY on the opposite side of AB

3. Cut 4 equal parts on AX and 3 equal parts on BY such that: AX₁ = X₁X₂ = X₂X₃ = X₃X₄ and BY₁ = Y₁Y₂ = Y₂Y₃

4. Join X₄Y₃ which intersects AB at P

∴  AP/PB = 4/3

Justification:

In ΔAPX₄ and ΔBPY₃, we have

∠APX₄ = ∠BPY₃ {Because they are vertically opposite angles}

∠PAX₄ = ∠PBY₃ {Because they are alternate interior angle}

ΔAPX₄ ΔBPY₃ {Because AA similarity}

∴  PAPB = AX₄/BY₃ = 4/3 {Because of C.P.C.T}

 

​Question: 3

Divide a line segment of length 14 cm internally in the ratio 2 : 5. Also, give justification for the construction.

Solution:

Steps of construction:

1. Draw a line segment AB of 14 cm

2. Through the points A and B, draw two parallel lines AX and BY on the opposite side of AB

3. Starting from A, Cut 2 equal parts on AX and starting from B, cut 5 equal parts on BY such that: AX₁ = X₁X₂ and BY₁ = Y₁Y₂ = Y₂Y₃ = Y₃Y₄ = Y₄Y₅

4. Join X₂Y₅ which intersects AB at P

∴  AP/PB = 2/5

Justification:

In ΔAPX₂ and ΔBPY₅, we have

∠APX₂ = ∠BPY₅ {Because they are vertically opposite angles}

∠PAX₂ = ∠PBY₅ {Because they are alternate interior angles}

Then, ΔAPX₂ ΔBPY₅ {Because AA similarity}

∴  AP/PB = AX₂/BY₅ = 2/5 {Because of C.P.C.T}