1.

D and E are points on the sides AB and AC respectively of a ∆ABC. In each of the following cases, determine whether DE║BC or not.(i) AD = 5.7cm, DB = 9.5cm, AE = 4.8cm and EC = 8cm. (ii) AB = 11.7cm, AC = 11.2cm, BD = 6.5cm and AE = 4.2cm. (iii) AB = 10.8cm, AD = 6.3cm, AC = 9.6cm and EC = 4cm. (iv) AD = 7.2cm, AE = 6.4cm, AB = 12cm and AC = 10cm.

Answer»

(i) We have: 

AD/DE = 5.7/9.5 = 0.6  

AE/EC = 4.8/8 = 0.6  

Hence, AD/DB = AE/EC 

Applying the converse of Thales’ theorem, 

We conclude that DE || BC. 

(ii) We have: 

AB = 11.7 cm, DB = 6.5 cm 

Therefore, 

AD = 11.7 -6.5 = 5.2 cm 

Similarly, 

AC = 11.2 cm, AE = 4.2 cm 

Therefore, 

EC = 11.2 – 4.2 = 7 cm 

Now, 

AD/DB = 5.2/6.5 = 4/5 

AE/EC = 4.2/7 

Thus, AD/DB ≠ AE/EC 

Applying the converse of Thales’ theorem, 

We conclude that DE is not parallel to BC. 

(iii) We have: 

AB = 10.8 cm, AD = 6.3 cm 

Therefore,

DB = 10.8 – 6.3 = 4.5 cm 

Similarly, 

AC = 9.6 cm, EC = 4cm 

Therefore, 

AE = 9.6 – 4 = 5.6 cm 

Now, 

AD/DB = 6.3/4.5 = 7/5 

AE/EC = 5.6/4 = 7/5 

⟹ AD/DB = AE/EC 

Applying the converse of Thales’ theorem, 

We conclude that DE ‖ BC. 

(iv) We have : 

AD = 7.2 cm, AB = 12 cm 

Therefore, 

DB = 12 – 7.2 = 4.8 cm 

Similarly, 

AE = 6.4 cm, AC = 10 cm 

Therefore, 

EC = 10 – 6.4 = 3.6 cm 

Now, 

AD/DB = 7.2/4.8 = 3/2 

AE/EC = 6.4/3.6 = 16/9 

This, AD/DB ≠ AE/EC 

Applying the converse of Thales’ theorem, 

We conclude that DE is not parallel to BC.


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