Question

prove:- If a line parallel to a side of a triangle in tersects the remaing sides in two distinct points, then the line divides the side in the same proportion.​

Answer 1

Answer:

Given: The theorem: if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points , then the other two sides are divided in the same ratio.

To find: Prove the theorem (Basic Proportionality Theorem)

Solution:

First we need to do some constructions.

In triangle PQR, let l be a line parallel to QR, let the intersected points by l be M on PQ and N on PR, and now join QN and MR.

Now After looking to the figure, we get:

               area( tri MPN ) / area ( tri NQM ) = PM / MQ ( i )

As both triangles have the height same (MN) and a common vertex (M).

Similarly :

          area( tri MPN ) / area ( tri NRM ) = PN / NR  (ii)

Now:

          area( tri NQM ) = area ( tri NRM ) iii)

          (triangle lies in parallel lines and have same base)

Now from equation i, ii and iii, we have:

          area( tri MPN )/  area ( tri NQM ) = area( tri MPN ) = area ( tri NRM )

So similarly:

          PM / MQ =  PN / NR

          Hence proved.