Question

If O is any point in the interior of triangle abc, then which of the following is true?
a) (OA+OB+OC)>(AB+BC+CA)
b) (OA+ OB+OC)>1/2(AB+BC+CA)
c) (OA+OB+OC)<1/2(AB+BC+CA)
d) (OA+OB+OC)=1/2(AB+BC+CA)

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{O is any point in the interior of trangle ABC}$

$\underline{\textsf{To find:}}$

$\textsf{Correct option from the alternatives}$

$\underline{\textsf{Solution:}}$

$\mathsf{We know that,}$

$\textsf{Sum of any two sides of a triangle is always greater than the third side}$

$\textsf{We have}$

$\mathsf{In\;\triangle\;OAB,\;AB\;<\;OA+OB}$.........(1)

$\mathsf{In\;\triangle\;OBC,\;BC\;<\;OB+OC}$........(2)

$\mathsf{In\;\triangle\;OAC,\;AC\;<\;OA+OC}$........(3)

$\mathsf{Adding,\;(1),(2)\;and\;(3),\;we\;get}$

$\mathsf{AB+BC+AC\;<\;OA+OB+OB+OC+OA+OC}$

$\mathsf{AB+BC+AC\;<\;2(OA+OB+OC)}$

$\mathsf{\dfrac{1}{2}(AB+BC+AC)\;<\;OA+OB+OC}$

$\implies\boxed{\mathsf{OA+OB+OC\;>\;\dfrac{1}{2}(AB+BC+AC)}}$

$\underline{\textsf{Answer:}}$

$\mathsf{Option\;(b)\;is\;correct}$