Question

] Prove the Theorem : 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

Answer:

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:

 <(△QMN) / <(△PMN) = MQ/ PM

(Both triangles have equal height with common vertex M)

∴<(△RMN)/ <(△PMN) = NR/PN ... (ii)

But <(△QMN) = <(△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

<(△QMN)/ <(△PMN)=<(△RMN)/ <(△PMN)

∴MQ/PM=NR/PN [henceproved]