## Chapter 4 triangles R.D. Sharma Solutions for Class 10th Math Exercise 4.2

**Exercise 4.2**

1. In ΔABC, D and E are points on the sides AB and AC respectively such that DE | | BC

(i) AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.

(ii) If AD/DB = 3/4 and AC = 15 cm, find AE

(iii) If AD/DB = 2/3 and AC = 18 cm, find AE

(iv) If AD = 4, AE = 8, DB = x-4, and EC = 3x – 19, find x.

(v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE.

(vi) If AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC .

(vii) If AD = 2 cm, AB = 6 cm and AC = 9 cm, find AE.

(viii) If AD/BD = 4/5 and EC = 2.5 cm, find AE .

(ix) If AD = x, DB = x – 2, AE = x+2 and EC = x – 1, find the value of x.

(x) If AD = 8x – 7, DB = 5x – 3, AE = 4x – 3 and EC = (3x -1), find the value of x.

(xi) If AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x – 3, find the value of x.

(xii) If AD = 2.5 cm, BD = 3.0 cm and AE = 3.75 cm, find the length of AC .

**Solution**

(i)

(ii)

(iii)

(iv)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

2. In a ∆ABC, D and E are points on the sides AB and AC respectively. For each of the

following cases show that DE | | BC:

(i) AB = 2cm, AD = 8cm, AE = 12 cm and AC = l8cm.

(ii) AB = 5.6cm, AD = 1.4cm, AC = 7.2 cm and AE = 1.8 cm.

(iii) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

(iv) AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm.

(i)

(xii)

2. In a ∆ABC, D and E are points on the sides AB and AC respectively. For each of the

following cases show that DE | | BC:

(i) AB = 2cm, AD = 8cm, AE = 12 cm and AC = l8cm.

(ii) AB = 5.6cm, AD = 1.4cm, AC = 7.2 cm and AE = 1.8 cm.

(iii) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

(iv) AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm.

**Solution**(i)

(ii)

(iii)

(iv)

3. In a ∆ABC, P and Q are points on sides AB and AC respectively, such that PQ || BC. If AP= 2.4 cm, AQ = 2 cm, QC = 3 cm and BC = 6 cm, find AB and PQ.

**Solution**

4. In a ∆ABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 2.4cm, AE = 3.2 cm, DE = 2cm and BC = 5 cm, find BD and CE.

Solution

5. In below Fig., state if PQ || EF.

**Solution**

6. M and N are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether MN || QR

(i) PM = 4cm, QM = 4.5 cm, PN = 4 cm and NR = 4.5 cm

(ii) PQ = 1.28 cm, PR = 2.56 cm, PM = 0.16 cm, PN = 0.32

**Solution**

(i)

(ii) For MN | | QR, by Theles theorem the required condition is :

PM/PQ = PN/PR

PM/PQ = 0.16/1.28 = (16/100)/(128/100)

= 16/128 = 1/8

PN/PR = 0.32/2.56 = 32/256 = 1/8

PM/PQ = PN/PR

Therefore MN | | QR.

7. In three line segments OA, OB, and OC, points L, M, N respectively are so chosen that LM || AB and MN || BC but neither of L, M, N nor of A, B, C are collinear. Show that LN || AC.

8. If D and E are points on sides AB and AC respectively of a ∆ABC such that DE || BC and BD = CE. Prove that ∆ABC is isosceles.

(ii) For MN | | QR, by Theles theorem the required condition is :

PM/PQ = PN/PR

PM/PQ = 0.16/1.28 = (16/100)/(128/100)

= 16/128 = 1/8

PN/PR = 0.32/2.56 = 32/256 = 1/8

PM/PQ = PN/PR

Therefore MN | | QR.

7. In three line segments OA, OB, and OC, points L, M, N respectively are so chosen that LM || AB and MN || BC but neither of L, M, N nor of A, B, C are collinear. Show that LN || AC.

**Solution**8. If D and E are points on sides AB and AC respectively of a ∆ABC such that DE || BC and BD = CE. Prove that ∆ABC is isosceles.

**Solution**