# 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:**

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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 AX_{1}=X_{1}X_{2}and BX_{1}=Y_{1}Y_{2}=Y_{2}Y_{3}

Step 4.Join X2Y3 which intersects AB at P

Justification:In ∆AX

_{2}P and ∆BY_{3}P, we have∠APX

_{2}=∠BPY_{3 }{Because they are vertically opposite angle}∠X

_{2}AP=∠Y_{3}BP∆AX

_{2}P and ∆BY_{3}P {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 AX_{1}=X_{1}X_{2}=X_{2}X_{3}=X_{3}X_{4}and BY_{1}=Y_{1}Y_{2}=Y_{2}Y_{3}

Step 4.Join x_{4}y_{3}which intersects AB at P.Therefore,

Justification:In ∆PX

_{4}and ∆BPY_{3}, we have∠APX

_{4}=∠BPY_{3 }{Because they are vertically opposite angles}∠PAX

_{4}=∠PBY_{3}{Because they are alternate interior angle}∆APX

_{4}∆BPY_{3}{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 ∆PX

_{2}and ∆BPY_{5}, we have∠APX

_{2}=∠BPY_{5}(vertically opposite angles)∠PAX

_{2}=∠PBY_{5}(Because they are alternate interior angle)∆APX

_{2 }∆BPY_{5}(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 x_{4}y_{5}which intersects AB at P.

Justification:In ∆PX

_{4}and ∆BPY_{5}, we have∠APX

_{4}=∠BPY_{5}(vertically opposite angles)∠PAX

_{4}=∠PBY_{5}(alternate interior angle)∆APX

_{4}∆BPY_{5}(AA similarity)Therefore,

(Because of C.P.C.T)