A triangle has sides with lengths: 2, 9, 1. How do you find the area of the triangle using Heron's formula?
1 Answer
There is no such triangle, but...
Explanation:
As noted by other contributors, it is not possible to form a triangle with sides
{(a+b > c), (b+c > a), (c+a > b):}
and these conditions are sufficient.
For fun, let us try to apply Heron's formula and see what happens.
The semi-perimeter
s = 1/2(a+b+c) = 1/2(2+9+1) = 6
Then Heron's formula tells us that the area is:
sqrt(s(s-a)(s-b)(s-c)) = sqrt(6(6-2)(6-9)(6-1))
color(white)(sqrt(s(s-a)(s-b)(s-c))) = sqrt(6(4)(-3)(5))
color(white)(sqrt(s(s-a)(s-b)(s-c))) = sqrt(-360)
i.e. not a real area.
Notice that the three conditions we wrote down above hold if and only if all of
Simply put, Heron's formula applies and gives a positive real value if and only if the conditions for sides of length
