Question

Take any point $O$ in the interior of a triangle $PQR$.

Is

$\left(i\right)OP+OQ>PQ$?

$\left(ii\right)OQ+OR>QR$?

$\left(iii\right)OR+OP>RP$?

Answer 1

If $O$ is a point in the interior of a given triangle, then three triangles $\Delta OPQ,\Delta OQR$, and $\Delta ORP$ can be constructed. In a triangle, the sum of the lengths of either two sides is always greater than the third side.

(i) Yes,

as $\Delta OPQ$ is a triangle with sides $OP,OQ$, and $PQ$.

$OP+OQ>PQ$

(ii) Yes,

as $\Delta OQR$ is a triangle with sides $OR,OQ$, and $QR$.

$OQ+OR>QR$

(iii) Yes, as $\Delta ORP$ is a triangle with sides $OR,OP$, and $PR$.

$OR+OP>PR$.