ANSWER:

Converse of basic proportionality theorem :  

If a line divides any two SIDES of a triangle in the same ratio then the line must be parallel to the third side.

SOLUTION :

1)  Given : D and E are the points on sides AB and AC. AB = 12 cm, AD = 8 cm, AE = 12 cm, and AC = 18 cm.

To prove : DE || BC.

DB= AB - AD

DB = 12 - 8

DB = 4 cm

EC = AC - AE

EC = 18 - 12

EC = 6 cm

In ∆ABC,

AD / DB = 8/4 = 2

And,  AE/EC = 12/ 6 = 2

so, AD / DB = AE/EC

Hence, DE || BC.

[By Converse of basic proportionality theorem]

2)  Given : D and E are the points on sides AB and AC.  AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm, and AE = 1.8 cm.

To prove : DE || BC.

DB= AB - AD

DB = 5.6 - 1.4

DB = 4.2 cm

EC = AC - AE

EC = 7.2 - 1.8

EC = 5.4 cm

In ∆ABC,

AD / DB = 1.4 /4.2 = 1/3

And,  AE/EC = 1.8/ 5.4 = 1/3

so, AD / DB = AE/EC

Hence, DE || BC.

[By Converse of basic proportionality theorem]

3) Given : D and E are the points on sides AB and AC.   AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm, and AE = 2.8 cm

To prove : DE || BC.

AD = AB – DB  

AD = 10.8 – 4.5

AD = 6.3 cm

EC = AC – AE  

EC = 4.8  – 2.8  

EC = 2 cm

In ∆ABC,

AD / DB = 6.3/4.5 = 7/5  = 1.4

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