Question

Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

Answer 1

Given, In ΔABC

D and E are the midpoints of AB and AC respectively, such that,

AD = BD and AE = EC.

We have to prove that: DE || BC.

Since, D is the midpoint of AB

∴ AD = DB

⇒ `(AD)/(BD)`  = 1  .......(i)

Also given, E is the midpoint of AC.

∴ AE = EC

⇒ `(AE)/(EC)` = 1

From equation (i) and (ii), we get,

`(AD)/(BD) = (AE)/(EC)`

By converse of Basic Proportionality Theorem,

DE || BC

Hence, proved.