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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