Class 10 RD Sharma Solutions – Chapter 11 Constructions – Exercise 11.1

Question 1. Determine a point that divides a line segment of length 12 cm internally in the ratio 2 : 3. Also, justify your construction.

Solution:

Steps of construction:

Step 1. Draw a line segment AB =12cm.

Step 2. Through the A and B draw two parallel line on each sides of AB.

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

Step 4. Join X2Y3 which intersects AB at P 

Justification: 

In ∆AX₂P and ∆BY₃P, we have

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

∠X₂AP=∠Y₃BP

∆AX₂P and ∆BY₃P {Because AA similarity}

Therefore,  {Because of C.P.CT}

Question 2. Divide a line segment of length 9 cm internally in the ratio 4 : 3. Also, give justification of the construction.

Solution:

Steps of construction:

Step 1. Draw a line segment AB = 9 cm.

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

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

Step 4. Join x₄y₃ which intersects AB at P.

Therefore, 

Justification:

In ∆PX₄ and ∆BPY₃, we have

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

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

 ∆APX₄     ∆BPY₃   {Because AA similarity}

Therefore,  {Because of C.P.C.T}

Question 3. Divide a line segment of length 14 cm internally in the ratio 2 : 5. Also, justify your construction.

Solution:

Steps of construction:

Step 1. Draw a line segment AB = 14 cm.

Step 2. Draw a ray AX making an acute angle with AB.

Step 3. From B, draw another ray BY parallel to AX.

Step 4. From AX, cut off 2 equal parts and from B, cut off 5 equal parts.

Step 5. Join 2 and 5 which intersects AB at P.

P is the required point which divides AB in the ratio of 2 : 5 internally.

Justification:

In ∆PX₂ and ∆BPY₅, we have

∠APX₂=∠BPY₅         (vertically opposite angles)

∠PAX₂=∠PBY₅        (Because they are alternate interior angle)

∆APX₂     ∆BPY₅     (Because AA similarity)

Therefore,

(Because of C.P.C.T)

Question 4. Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5.

Solution:

Steps of construction:

Step 1. Draw a line of 8cm.

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

Step 3. Cut 4 equal parts AX and 3 equal parts on BY.

Step 4. Join x₄y₅ which intersects AB at P.

Justification:

In ∆PX₄ and ∆BPY₅, we have

∠APX₄=∠BPY₅         (vertically opposite angles)

∠PAX₄=∠PBY₅       (alternate interior angle)

∆APX₄     ∆BPY₅   (AA similarity)

Therefore,

(Because of C.P.C.T)