Q11 of 32 Page 6

Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio. (CBSE 2002, 2004, 2006, 2007, 2009, 2010)

Let a ∆ABC in which a line DE parallel to BC intersects AB at D and AC at E.

To prove DE divides the two sides in the same ratio.




Construction join BE, CD and draw EF ⊥ AB and DG ⊥ AC.



Proof Here,


[ area of triangle = × base × height]



Similarly,



Now,


Since,


∆BDE and ∆DEC lie between the same parallel DE and BC and on the same base DE.


So, area (∆BDE) = area(∆DEC) …..(iii)


From Equation (i), (ii) and (iii),


AD/DB = AE/EC


Hence proved.

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