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Question
AM is a median of a triangle ABC.
Is AB + BC + CA > 2 AM?
(Consider the sides of triangles ΔABM and ΔAMC.)
Sum
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Solution
In a triangle, the sum of the lengths of either two sides is always greater than the third side.
In ΔABM,
AB + BM > AM ...(i)
Similarly, in ΔACM,
AC + CM > AM ...(ii)
Adding equation (i) and (ii),
AB + BM + MC + AC > AM + AM
AB + BC + AC > 2AM
Yes, the given expression is true.
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