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Question

State and prove converse of BPT (basic proportionality theorem).


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Solution

Converse of Basic proportionality theorem (BPT): According to this theorem, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Suppose a line DE, intersects the two sides of a triangle AB and AC at D and E, such that;

ADDB=AEEC........eq1

Assume DE is not parallel to BC. Now, draw a line DE' parallel to BC.

Hence, by similar triangles,

ADDB=AE'E'C........eq2

From eq. 1 and 2, we get;

AEEC=AE'E'C

Adding 1 on both sides:

AEEC+1=AE'E'C+1AE+ECEC=AE'+E'CE'CACEC=ACE'C

So, EC=E'C

This is possible only when E and E' coincides.

But,DE'//BC

Therefore, DE//BC.

Hence, proved.


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