Question

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

Answer 1

Given: In ΔABC, DE || BC

To prove: `(AD)/(DB) = (AE)/(EC)`

Construction: Draw EM ⊥ AB and DN ⊥ AC.

Join B to E and C to D.

Proof: In ΔADE and ΔBDE

`(ar(ΔADE))/(ar(ΔBDE)) = (1/2 xx AD xx EM)/(1/2 xx DB xx EM) = (AD)/(DB)`   .....(i) `["Area of Δ" = 1/2 xx "base" xx "corresponding altitude"]`

In ΔADE and ΔCDE

`(ar(ΔADE))/(ar(ΔCDE)) = (1/2 xx AE xx DN)/(1/2 xx EC xx DN) = (AE)/(EC)`  .....(ii)

Since DE || BC  ...[Given]

∴ ar(ΔBDE) = (ΔCDE)  .....(iii) [Δ on the same base and between the same parallel sides are equal in area]

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

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

Hence proved.