CAT Basic-Proportionality Theorem or Thales' Theorem - Practice Questions & MCQ

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.

Given: in which and DE intersects AB and AC at D and E respectively.

To prove: 

Construction: Join BE and CD. Draw and .

Proof: We have

Now, and being on the same base DE and between the same parallels DE and BC, we have

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

 

Theorem (Converse of Thales Theorem): If a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.

Given:  and a line intersecting AB at D and AC at E, such that .

To Prove: 

Proof: 

If possible, let DE not be parallel to BC. Then, there must be another line through D which is parallel to BC.

Then, by Thales' theorem, we have

From (i) and (ii), we get

        

This is possible only when E and F coincide.

Hence,