Constructions Exercise 11.1 | Extra Documents, Videos & Tests for Class 10 PDF Download

Q.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

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 }

 { Because of C.P.C.T } 

Q.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

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 }

 { Because of C.P.C.T } 

Q.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:

(i) Draw a line segment AB of 14 cm

(ii) Through the points A and B, draw two parallel lines AX and BY on the opposite side of AB

(iii)  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

(iv) Join X₂Y₅ which intersects AB at P

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 }

 { Because of C.P.C.T }