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)

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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