Maths-
General
Easy

Question

Prove the following statement.
The length of any one median of a triangle is less than half the perimeter of the triangle.

Hint:

  • Triangle inequality theorem
  • According to this theorem, in any triangle, sum of two sides is greater than third side,
  • a < b + c
  • b < a + c
  • c < a + b
    • The line segment which joins the vertex of triangle to midpoint of opposite side is called the ‘median’.
    • Perimeter of triangle is sum of sides.
    • Perimeter = a + b + c

The correct answer is: Hence, proved.


    • The length of any one median of a triangle is less than half the perimeter of the triangle.

     
                                                  
    • Step 1:

    Let,

    Side AB = a

    Side BC = b

    Side AC = c

    And median AM be Ma

    • Step 2:

    In △ABM,

    AM is median so M is midpoint of BC

     BM = b over 2

    So, according to triangle inequality theorem,

    Ma < a + b over 2 ---- eq. 1

    In △ACM,

    AM is median so M is midpoint of BC

     MC = b over 2

    So, according to triangle inequality theorem,

    Ma < c + b over 2 ---- eq. 2

    Add eq. 1 and eq. 2.

    Ma + Ma < a + b over 2 + c + b over 2

    2Ma < a + b + c

    fraction numerator a plus b plus c over denominator 2 end fraction

    • Final Answer: 
    Hence, proved.

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