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Question 3
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.

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Solution

Consider ΔABC in which a line DE parallel to BC intersects AB at D and AC at E.
To prove that DE divides the two sides in the same ratio.
i.e., ADDB=AEEC


Construction Join BE, CD and draw EF AB and DG AC.
Proof
Here, ar(ΔADE)ar(ΔBDE)=12×AD×EF12×DB×EF [ area of triangle =12×base×height]
=ADDB ….(i)
Similarly, ar(ΔADE)ar(ΔDEC)=12×AE×GD12×EC×GD=AEEC …..(ii)
Now, since, ΔBDE and ΔDEC lie between the same parallel DE and BC and on the same base DE.
So, ar(ΔBDE)=ar(ΔDEC) ……(iii)
From Eq.s (i), (ii) and (iii), we get
ADDB=AEEC


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