Question

Why is length of median less than the average of two sides containing it

Answer 1

Answer:

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In the diagram, AM is a median of triangle ABC.

We want to see why AM < ( AB + AC ) / 2.

This is the same as  2 AM < AB + AC.

An easy way to see this is to make DCB as another copy of ABC (rotate 180° around M) so that ABCD is a parallelogram.

Then AB + AC = AB + BD

and 2 AM = AD (since diagonals of a parallelogram bisect each other).

Since the straight line is the shortest path between two points, we have

AD < AB + BD  (this is called the Triangle Inequality )

and this is the same as

2 AM < AB + AC.