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AM is a median of a triangle ABC. Is AB + BC + CA > 2 AM? (Consider the sides of triangles ∆ABM and ∆AMC.)

3.  AM is a median of a triangle ABC.
            Is \small AB+BC+CA> 2AM ?
(Consider the sides of triangles \small \Delta ABM and \small \Delta AMC.)

        

Answers (1)
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As we know that the sum of two sides of ANY triangle is always greater than the third side(Triangles Inequality Rule).

So,

In \small \Delta ABM :

\overline {AB}+\overline {BM}>\overline{AM}...........(1)

In \small \Delta AMC :

\overline {AC}+\overline {CM}>\overline{AM}...........(2)

Adding (1) and (2), we get

\overline{AB}+\overline {AC}+\overline{BM}+\overline {CM}>\overline{AM}+\overline{AM}

As we can see M is the point in line BC So, we can say

\overline{BM}+\overline {CM}=\overline {BC}

So our equation becomes

\overline{AB}+\overline {AC}+\left (\overline{BM}+\overline {CM} \right )>\overline{AM}+\overline{AM}

\small AB+BC+CA> 2AM.

Hence it is a True statement.

 

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