Question

Converse of BPT theorem​

Answer 1

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.

If AD AE

---- = ------ then DE || BC

DB EC

Given : A Δ ABC and a line intersecting AB in D and AC in E,  

such that AD / DB = AE / EC.

Prove that : DE || BC  

 

Let DE is not parallel to BC. Then there must be another line that is parallel to BC.

Let DF || BC.

Statements

Reasons

1) DF || BC 1) By assumption

2) AD / DB = AF / FC 2) By Basic Proportionality theorem

3) AD / DB = AE /EC 3) Given

4) AF / FC = AE / EC 4) By transitivity (from 2 and 3)

5) (AF/FC) + 1 = (AE/EC) + 1 5) Adding 1 to both side

6) (AF + FC )/FC = (AE + EC)/EC 6) By simplifying

7) AC /FC = AC / EC 7) AC = AF + FC and AC = AE + EC

8) FC = EC 8) As the numerator are same so denominators are equal

This is possible when F and E are same. So DF is the line DE itself.

∴ DF || BC  

Examples  

1) D and E are respectively the points on the sides AB and AC of a ΔABC such that AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm, prove that DE || BC.

Solution :  

 

AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm  

∴ BD = AB – AD = 5.6 – 1.4 = 4.2 cm  

And EC = AC – AE = 7.2 – 1.8 = 5.4 cm  

Now, AD / DB =1.4 / 4.2 = 1/3  

And AE / EC = 1.8 / 5.4 = 1/3  

⇒ AD / DB = AE / EC  

DE || BC ( by converse of basic proportionality theorem)

Step-by-step explanation:

Answer 2

Converse of BPT :

If a line drawn divides two sides of a triangle in the same ratio, then that line is parallel to the third side

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