Question

O is a point in the exterior of ∆ABC. What symbol '>', '<' or '=' will you use to complete the statement OA + OB .. AB? Write two other similar statements and shown that

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

Answer 1

Because the sum of any two sides of a triangle is always greater than the third side, in triangle OAB, we have:

$$\begin{gathered} OA+OB>AB...\left(\mathrm{i}\right) \\ OB+OC>BC...\left(\mathrm{ii}\right) \\ OA+OC>CA...\left(\mathrm{iii}\right) \\ \mathrm{On}\mathrm{adding}\mathrm{equations}\left(\mathrm{i}\right),\left(\mathrm{ii}\right)\mathrm{and}\left(\mathrm{iii}\right),\mathrm{we}\mathrm{get}: \\ OA+OB+OB+OC+OA+OC>AB+BC+CA \\ 2(OA+OB+OC)>AB+BC+CA \\ OA+OB+OC>\frac{AB+BC+CA}{2} \end{gathered}$$