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 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 = 
So, according to triangle inequality theorem,
Ma < a + ---- eq. 1
In △ACM,
AM is median so M is midpoint of BC
MC =
So, according to triangle inequality theorem,
Ma < c + ---- eq. 2
Add eq. 1 and eq. 2.
Ma + Ma < a + + c +
2Ma < a + b + c
- Final Answer:
Hence, proved.
---- eq. 1 ---- eq. 2Add eq. 1 and eq. 2.
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