Q) 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.

Ans: 

Given:

Let D and E be two distinct points on AB and AC respectively such that DE ǁ BC.

Construction:

Let two points M and N be on AD and AE. Join ME and DN such that ME ⊥ AD and DN ⊥ AE.

Join DC and BE

To prove: 

Proof:

In Δ ADE and Δ BDE,

=

=              —-(1)

Similarly, in Δ AED and Δ CED,

=

=              —-(2)

Next, in Δ BDE and Δ CED:

∵  Area of triangles between two same parallel sides with same base are equal]

∴ Area Δ BDE = area Δ CED           —-(3)

Next, from equation 1 and equation 3; we get:

     (from equation 3)

By putting values from equation 1 and equation 2, we get:

Hence Proved !

Note: This is proving of Basic Proportionality Theorem (BPT). Read it carefully !