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