Question related to triangles.

Question

I am stuck at a question:

O is a point in the interior of ∆PQR , then which of the following is true:

1)$(OP+OQ+OR)<1/2(PQ+QR+PR)$

2)$(OP+OQ+OR)=1/2(PQ+QR+PR)$

3)$(OP+OQ+OR)>1/2(PQ+QR+PR)$

Please can someone explain me how to do this?

Thanks for the help.

score 1 · Accepted Answer · 2015-06-18T08:43:12.597

1

enter image description here I will use the following fact to prove your result:

Fact: In a triangle sum of two sides is always greater than 3rd side

So $PQ~<~OP+OQ$

$PR~<~OP+OR$

$QR~<~OQ+OR$

Now adding these equations we have

$PQ+PR+QR<2(OP+OQ+OR)$

So $(OP+OQ+OR)>\frac{1}{2}(PQ+PR+QR)$

edited Jun 18 '15 at 08:43

answered Jun 18 '15 at 07:29

Wait...... Shouldn't it be this: $PQ < OP+OQ$ , $PR < OP+OR$ , $QR < OQ+OR$ – codetalker Jun 18 '15 at 07:36
yes ! by mistake i wrote wrongly – Jun 18 '15 at 08:43

1 Answers1