# Ex.6.4 Q3 The-Triangle-and-its-Properties Solutions-NCERT Maths Class 7

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## Question

$$\rm{}AM$$ is a median of a triangle $$\rm{}ABC.$$ Is $$\rm{}AB + BC + CA > 2 AM?$$ (Consider the sides of triangles $$\Delta{ABM}$$ and $$\Delta{AMC}$$.) ## Text Solution

What is known?

ABC is a triangle and AM is a median of triangle ABC.

What is unknown?

Is $$AB + BC + CA > 2\, \rm AM?$$

Reasoning:

In this question it is asked if $$\rm{}AB + BC + CA > 2 AM$$ or not. This question is also based on the property that the sum of lengths of two sides of a triangle is always greater than the third side. In such kind of problems, you just visually identify the triangle $$ABC$$ and $$AM$$ is the median which further divides triangle $$\rm{}ABC$$ into two more triangles i.e. triangle $$\rm{}ABM$$ and $$\rm{}AMC.$$ Now consider any two sides of first triangle $$ABM$$ and use the above property then consider the any two sides of another triangle $$\rm{}AMC$$ and use the above property. Now, add $$\rm{}L.H.S.$$and $$\rm{}R.H.S.$$ of both the triangles.

Steps:

In triangle $$\rm{}ABM,$$

$$AB + BM > AM{\rm{ }} \qquad \ldots \ldots \ldots \ldots \ldots \ldots .{\rm{ }}\left( 1 \right)$$

In triangle $$\rm{}AMC,$$

$$AC + MC > AM{\rm{ }} \qquad \ldots \ldots \ldots \ldots \ldots \ldots .{\rm{ }}\left( 2 \right)$$

Adding equation $$\rm{}(1)$$ and $$\rm{}(2)$$ we get,

\begin{align}&\rm{}AB + \rm{}BM + \rm{}AC + \rm{}MC > AM + AM\\&\rm{}AB + AC + BM + MC > 2AM\\&\rm{}AB + AC + BC > 2AM\end{align}

Hence, it is true

Useful Tip:

Whenever you encounter problems of this kind, it is best to think of the property based on sum of lengths of any two sides of a triangle is always greater than the third side

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