Question

Take any point O in the interior of a triangle PQR. Is
(i) OP + OQ > PQ?
(ii) OQ + OR > QR?
(iii) OR + OP > RP?

Answer 1

If we take any point O in the interior of a triangle PQR and join OR, OP, OQ.
Then, we get three triangles ΔOPQ, ΔOQR and ΔORP is shown in the figure below.



We know that,
The sum of the length of any two sides is always greater than the third side.
(i) Yes, ΔOPQ has sides OP, OQ and PQ.
So, OP + OQ > PQ
(ii) Yes, ΔOQR has sides OR, OQ and QR.
So, OQ + OR > QR
(iii) Yes, ΔORP has sides OR, OP and PR.
So, OR + OP > RP