Theorem 6.1 Basic proportionality theorem (BPT) : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio Class 10

Given : △ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively.

To prove : AD/DB = AE/EC

Construction : Join BE and CD and draw DM and EN perpendicular to AC and AB respectively.

Proof : area of △ADE = ½ × base × height 

So, ar(ADE) = ½ × AD × EN

Similarly, ar(BDE) = ½ × BD × EN

ar(ADE) = ½ × AE × DM and ar(DEC) = ½ × EC × DM

Note that △BDE and △DEC are on the same base DE and between the same parallels BC and DE.

So, ar(BDE) = ar(DEC)…….(iii)

Therefore, from (i), (ii) and (iii), we have :

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