Question

Prove that : If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion.Prove that : If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion.​

Answer 1

Given: In a △PQR, line l∥ side QR, line l intersect the sides PQ and PR in two distinct points M and N respectively.

To prove: MQPM=NRPN ... (i)

Construction: segQN and segRM are drawn.

Proof: A(△QMN)A(△PMN)=MQPM

(Both triangles have equal height with common vertex M)

∴A(△RMN)A(△PMN)=NRPN ... (ii)

But A(△QMN)=A(△RMN), because they are between parallel lines MN and QR and have equal height corresponding to their common base MN ..... (iii)

From (i), (ii) and (iii), we get

A(△QMN)A(△PMN)=A(△RMN)A(△PMN)

∴MQPM=NRPN [henceproved]