Question

If O is any point in the interior of a triangle ABC, then $OA+OB+OC>\frac{1}{2}(AB+BC+CA)$

• A. True
• B. False

Answer 1

The correct option is A True

The statement is true.

Proof:

In a triangle, sum of any two sides is greater than the third side.

In triangle $OAB$ ,

$OA+OB>AB$ ............. (1)

In triangle $OBC$ ,

$OB+OC>BC$ ............. (2)

In triangle $OAC$ ,

$OA+OC>AC$ ............. (3)

Add (1), (2) and (3)

$2(OA+OB+OC)>AB+BC+AC$

$\Rightarrow OA+OB+OC>\frac{1}{2}(AB+BC+AC)$