Can we Prove that ……..
The length of any side of a triangle is not more than half of its perimeter
Key Concepts
Triangle Inequality
Perimeter
Geometry
Check the Answer
Answer: Yes we can definitely prove that by Triangle Inequality
Mathematical Circles – Chapter 6 – Inequalities Problem 3
Mathematical Circles by Dmitri Fomin , Sergey Genkin , Llia Itenberg
Try with Hints
We can start this sum by using this picture below
The length of the three sides of this triangle are a,b and c. So if we apply triangle inequality which implies that the length of one side of a triangle is less than the sum of the lengths of the two sides of that triangle. In reference to the theorem
b + c > a
So can you try to do the rest of the sum ????????
According to the question we have to find the perimeter at first
Perimeter is the sum of the length of all sides of the triangle = a + b + c
And the length of each side is a or b or c.
We have to prove : a + b + c > length of any one side
This can be one of the most important hint for this problem. Try to do the rest of the sum …………………………..
Here is the rest of the sum :
As stated above if we use triangle inequality :
b + c > a
Lets add a to both the sides
a + b + c > a + a
a + b + c > 2 a
The left hand side of the above inequality is the perimeter of this triangle.
perimeter > 2 a
So , \(\frac {perimeter}{2} > a \)
\(\frac {perimeter}{2} \) = semi perimeter
Hence this is proved that the length of one side of a triangle is less than half of its perimeter.