Question

if Ois the point in the exterior of triangle ABC show that 2(oa+ob+oc)>ab+bc+ca
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Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

O is any point in the exterior of triangle ABC.

Now,

$\rm \: In \: \triangle \: OAB$

We know that, sum of any two sides of a triangle is greater than third side.

So, using this, we get

$\rm\implies \: \boxed{\sf{ \:\rm \: \:OA + OB > AB \: }} - - - (1)$

Now,

$\rm \: In \: \triangle \: OAC$

We know that, sum of any two sides of a triangle is greater than third side.

So, using this, we get

$\rm\implies \:\boxed{\sf{ \:\rm \: OA + OC > AC \: }} - - - (2)$

Now,

$\rm \: In \: \triangle \: OBC$

We know that, sum of any two sides of a triangle is greater than third side.

So, using this, we get

$\rm\implies \:\boxed{\sf{ \:\rm \: \: OB + OC > BC \: \: }} - - - (3)$

So, on adding equation (1), (2) and (3), we get

$\rm \: OA + OB + OA + OC + OB + OC > AB + AC + BC$

$\rm\implies \:2(OA + OB + OC) > AB + BC + CA$

Hence, Proved

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Additional Information :-

1. The sum of all interior angles of a triangle is supplementary.

2. The sum of all exterior angles of a triangle is 360°.

3. Angle opposite to longest side is always greater.

4. Side opposite to greater angle is always longest.

5. The exterior angle of a triangle is equals to sum of interior opposite angles.

Answer 2

Step-by-step explanation:

We know that in a triangle, sum of two sides is greater than the third side. [Triangle inequality theorem] Using this in the above figure, Let O is a point in the exterior of $\triangle A B C$

To prove:

$\tt \: 2(O A+O B+O C)>A B+ B C+A C$

Proof: In $\triangle A O B$,

$\tt \: O A+O B>A B[ \text{Using Triangle inequality theorem} ] \ldots (i)$

In $\triangle B O C$,

$\text{O B+O C>B C}[ \text{ Using Triangle inequality theorem}] \ldots \ \tt (ii)$

In $\triangle A O C$,

$\tt O A+O C>A C[ \text{Using Triangle inequality theorem}] \ldots (iii)$

Adding (i), (ii) and (iii) we get,

$\text{\( \tt \: O A+O A+O B+O B+O C+O C> \) \( \tt A B+A C+B C \)}$

$\text{\( \tt \Longrightarrow 2(O A+O B+O C)>A B+B C \)}$