Question

If a line drawn parallel to one side of a triangle intersecting the other two sides in distinct points divides the two sides in the same ratio, then it is parallel to the third side.

State and prove the converse of the above statement.

Answer 1

Converse: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same proportion.

Given: ΔABC, R || BC intersects AB and AC at D and E, respectively.

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

Construction: Draw EF ⊥ AB and DG ⊥ AC

Join CD, BЕ

Proof: `(ar(ADE))/(ar(BDE)) = (1/2 xx AD xx EF)/(1/2 xx BD xx EF) = (AD)/(BD)`    ...(i)

`(ar(ADE))/(ar(CDE)) = (1/2 xx AE xx DG)/(1/2 xx CE xx DG) = (AE)/(CE)`   ....(ii)

ar(BDE) = ar(ADE)    ...(iii)

Triangles that lie on the same base and between the same parallel lines have equal areas.

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

⇒ `(AD)/(BD) = (AE)/(EC)`