Question
Fri July 06, 2012 By:

'O' is any point in the interior of triangle ABC.then which of the following is true and how? A) (OA+OB+OC)>(AB+BC+CA) B) (OA+OB+OC)> 1/2 (AB+BC+CA) c) (OA+OB+OC)

Expert Reply
Sun July 08, 2012
 
Answer : Given :O' is any point in the interior of triangle ABC
To find : 
which of the following is true and how? A) (OA+OB+OC)>(AB+BC+CA)
B) (OA+OB+OC)> 1/2 (AB+BC+CA)
C) (OA+OB+OC)< 1/2 (AB+BC+CA)
D) None of these


The left inequality follows from three applications of the Triangle Inequality. we have
AB < OA + OB
BC < OB + OC
CA < OC + OA
 
Adding all these together, we get
(AB+BC+CA)< 2(AO + BO + CO) as required.
 
For the right inequality we need triangle law of property, we get
OA + OB < CA + CB
OB + OC < AB + AC
OC + OA < BC + BA
Adding them all together gives 2(AO + BO + CO) < 4 (AB+BC+CA)
=> (AO + BO + CO) < 2 (AB+BC+CA)
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