Basic proportionality theorem

FOR CLASS 8th

1. BASIC PROPORTIONALITY THEOREM Class-X

2. STATEMENT If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct point, the other two sides are divided in the same ratio.

3. PROOF We are given a triangle ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively. W e need to prove that AD/DB = AE/EC Let us join BE and CD and then draw DM ┴ AC and EN ┴ AB Now area of ∆ADE (1/2 x base x height) = 1/2 x AD x EN So, ar (ADE) = ½ AD x EN Similarly , ar (BDE) = ½ BD x EN ar (ADE) = ½ AE x DM ar (DEC) = ½ EC x DM

4. Therefore , ar (ADE)/ ar (BDE) = ½ AD x EN / ½ BD x EN = AD/BD ………………………(1) and , ar (ADE)/ ar (DEC) = ½ AE x DM / ½ EC x DM = AE/EC …………………………….(2) Note that ∆BDE and ∆DEC are on the same base DE and between the same parallels BC and DE So, ar (ADE) = ar (DEC) …………………………………..(3) Therefore from (1),(2)and (3), we have , AD/BD = AE/EC Hence proved.