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 \(PQ\).
\(OP + OQ > PQ\)

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

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