Question

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.

Answer 1

Let a ΔABC in which a line DE parallel to SC intersects AB at D and AC at E.

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

i.e, `(AD)/(DB) = (AE)/(EC)`

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

Proof: Here `(ar(ΔADE))/(ar(ΔBDE)) = (1/2 xx AD xx EF)/(1/2 xx DB xx EF)`   .......[∵ Area of triangle = `1/2` × base × height]

= `(AD)/(DB)`  .......(i)

Similarly, `(ar(ΔADE))/(ar(ΔDE)) = (1/2 xx AE xx GD)/(1/2 xx EC xx GD) = (AE)/(EG)`   ......(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 equations (i), (ii) and (iii),

`(AD)/(DB) = (AE)/(EC)`

Hence proved.