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

Last updated date: 23rd Mar 2023

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Answer

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Hint: Here, we will use the property of triangle i.e.., the sum of lengths of any two sides in a triangle should be greater than the length of the third side to prove the given condition $AB + BC + CA > 2AM$.

Complete step-by-step answer:

Given,

AM is a median of a triangle ABC and the triangle is divided into$\Delta ABM{\text{ }}$and $\Delta AMC$.

As we know the property of a triangle i.e.., the sum of lengths of any two sides in a triangle should be greater than the length of the third side. Therefore, let us consider the triangle ABM, we get

$AB + BM > AM \to (i)$

Similarly, from$\Delta AMC$, we get

$AC + MC > AM \to (ii)$

Let us add equation (i) and (ii), we get

$\Rightarrow AB + BM + AC + MC > AM + AM \\$

$\Rightarrow AB + AC + (BM + MC) > 2AM \to (iii) \\ $

From the $\Delta ABC$, we know that

$\Rightarrow$ $BM + MC = BC \to (iv)$

So, let us substitute the equation (iv) in equation (iii), we get

$\Rightarrow$ $AB + BC + CA > 2AM$

Hence, equation (i) is proved.

Note: A median of a triangle is a line segment that joins a vertex to the midpoint of the side that is opposite to the vertex bisecting into two equal parts. Here, $BM = MC$ and the area of the triangles ABM and AMC are equal.

Complete step-by-step answer:

Given,

AM is a median of a triangle ABC and the triangle is divided into$\Delta ABM{\text{ }}$and $\Delta AMC$.

As we know the property of a triangle i.e.., the sum of lengths of any two sides in a triangle should be greater than the length of the third side. Therefore, let us consider the triangle ABM, we get

$AB + BM > AM \to (i)$

Similarly, from$\Delta AMC$, we get

$AC + MC > AM \to (ii)$

Let us add equation (i) and (ii), we get

$\Rightarrow AB + BM + AC + MC > AM + AM \\$

$\Rightarrow AB + AC + (BM + MC) > 2AM \to (iii) \\ $

From the $\Delta ABC$, we know that

$\Rightarrow$ $BM + MC = BC \to (iv)$

So, let us substitute the equation (iv) in equation (iii), we get

$\Rightarrow$ $AB + BC + CA > 2AM$

Hence, equation (i) is proved.

Note: A median of a triangle is a line segment that joins a vertex to the midpoint of the side that is opposite to the vertex bisecting into two equal parts. Here, $BM = MC$ and the area of the triangles ABM and AMC are equal.

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