Chapter 4 triangles R.D. Sharma Solutions for Class 10th Math Exercise 4.2
Exercise 4.21. 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)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2LYPH8HhgdSf-ctME8FpMPqNySuPe9Mzo84aXbrJN8slWNxiCkmdluKJl6DWYnB1TM9OqkV9ZN6vpw60p77HID8CD8RdCvSLmX7wk6Wa8Is2cJSWBQK-yOfZ-HVqnyYodMAUAcceeZ4I/s1600-rw/1+ii+img.PNG)
(iii)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimLmwniQBlPu_tO7Eqz7U_wfPGOW7v_5yYGmcleHXZIswurk76UTRvGAHhbBF2EV0YEfr3kLUrtozCGJLNw1joX5y93ype4C2Q0FCJTzy1-bCWQVXlhcG1WWqIXoK5XId95r06dbrwnrI/s1600-rw/1+iii+img.PNG)
(iv)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK1T3gCxCIxdPofXNkZhMKgKawk-vUFEuWwCrJwzGJgpA1ubVyTzhmaW7Uo-SaRzaPLwPvSMSvkz3a7JUuve96e4DN6A2vQWpZZPXnKG5tIsWd_ocbTEQ0hVESGFs5MUOtnq_6ccTegVE/s400-rw/1+iv.PNG)
(iv)
(v)
(vi)
(vii)
(viii)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-7vXZHVPKmGZCnEaQTW98qw8ODnzU0iEca59qXo7eFRiWhwwKqOCe63zoKstotEMczmIpEEa9OftKfPvDucbRMoWhE4sbaxLRb8uDdFxr7lZ-7tpFgO8xJLSjRHfOJqT7i9lD1t-WwrQ/s1600-rw/1+viii.PNG)
(ix)
(x)
(xi)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzdAllRMXnjsAjLt-gQq6ujfgAnYfLpokCkc9j-pzkeZ6dL8Mu1TQ_ubm5nzcV0QNDwK8Grn7kIwkSXdQTENTb9DfV8p1sG-ySZ4_oZLoWAAsGgDrscbnmiFFyfTaCzKGxMehoz6MQqK0/s1600-rw/1+xi.PNG)
(xii)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHE5a6gfbVVGvp1e7IobTsavIFIANm2MkQyu87bE3EEU3d-N_OZAu7ZqvGY952s149vgtowt7AbIYyiWOT3cYFLmdQ-CyxkGUQhSLPzBXZnoZU6DNGXEeo6AK2IKUcLXUlNaUxtpHUYH8/s1600-rw/1+xii.PNG)
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)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg21gLv0or-tYphpwl09pHfYmTLxB1GKIjZTx_WiIZzA6Azhk_YJpRAZJs60wgUWYyesuFYaZkArsT-HksdJr_egKqMFuC6jn3-bS3WBoOnBkNuZsJYoyRKnkaIu7jIVdGuFmDJANNGhs0/s400-rw/2+i.PNG)
(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)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY_XRZmGYtz02ahR8chJSSZLhJyAdorhss7IGkTbDTRoJw36fdk_r3v3_OcUFrLApASyPXEh1SdEJijsxsmw6RIF-bVB8IRJvNWXZ-q0-Ut47GEXdW_ZiwfJWhm5hf8VpzHwYAp37YCzs/s1600-rw/2+ii+a.PNG)
(iii)
(iv)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiULNO1dMx9cUQo5RGPOHeB2huqgMcDGtSNr3x9kixhzBMGr4rFhWsi_XhmWOMcL9CVTep0G8ApSJTrOnZ8EZVR1PwuUyW9pNLDcbRGmUllK-RLsZ7BE5bTNSludbkRs06w8z81FYRa1HQ/s640-rw/2+iv.PNG)
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
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSWr-zxUiHrkA_u0sMO_o50Qq8E0XgkK-Cv4TGsoy0xa4_e5zi0yQwgza7XhIOYsfRY62EQ3kgPNRJDZaEtS_j4WILxt0wp1XokKnTB7XGDzvUdCgCtj_kMVr2eix0Guhi8-uelt4hsYs/s1600-rw/5+ans.PNG)
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
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.
Solution
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhY2ugGDdbXVdjTQ8-ub-Q40Pj8KXBL01LdJEVsXBsbpehPDwKiH5o1qY4Y3XQWuaYjK7oT_YOssFHL5eMMgKUXaAT28PCCz8DEx9Ivjxbcl0ScTSTdmutEnkmSh0edmI8s-y95cZoV_u8/s1600-rw/7+img.PNG)
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
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMPmPqZEfB7-chLW88ZVg7opes_FGYQvkFu8GmYfqNTgCnyp8KdVmt2HKeKnlPm3UCirQ42PFjgYybS9MdnJOdFYJ3rCrV63UXx9ZEZHIRxee4ktHNZS-qf0tEaptisoig3kauvCVuflw/s1600-rw/8+img.PNG)
(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