Question

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

1. OP + OQ > PQ?
2. OQ + OR > QR?
3. OR + OP > RP?

Answer 1

If $O$ is a point in the interior of a given triangle, then three triangles $ΔOPQ, ΔOQR$, and $ΔORP$ 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 $ΔOPQ$ is a triangle with sides $OP, OQ,$ and $PQ$.
$OP + OQ > PQ$

---

(ii) Yes , as $ΔOQR$ is a triangle with sides $OR, OQ,$ and $QR$.
$OQ + OR > QR$

---

(iii) Yes , as $ΔORP$ is a triangle with sides $OR, OP$, and $PR$.
$OR + OP > PR$