# A triangle has sides with lengths: 6, 9, and 15. How do you find the area of the triangle using Heron's formula?

Area of triangle is zero

#### Explanation:

In general, a triangle will exist only when the sum of two sides is greater than the longest side.

The longest side of given triangle $15$ is equal to the sum of other two sides $6$ & $9$ thus triangle will be a straight line or triangle does not exist.

Now, the semi-perimeter $s$ of triangle with the sides $a = 6$, $b = 9$ & $c = 15$

$s = \setminus \frac{a + b + c}{2} = \setminus \frac{6 + 9 + 15}{2} = 15$

The area of triangle by using Herio's formula

$\setminus \Delta = \setminus \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

$= \setminus \sqrt{15 \left(15 - 6\right) \left(15 - 9\right) \left(15 - 15\right)}$

$= 0$

The area of the triangle will be zero

Jul 12, 2018

Triangle Inequality Theorem :

$\text{The sum of the lengths of any two sides of a triangle}$

$\text{must be greater than the length of the third side.}$

#### Explanation:

Let ,

$a = 6 , b = 9 \mathmr{and} c = 15$

=>6+9=15=>color(red)(a+b=c

So, according to Triangle Inequality Theorem ,

$\text{color(red)"it is not possible to construct a triangle.}$

OR

$\therefore \text{ Semi perimeter of triangle is :}$

$s = \frac{a + b + c}{2} = \frac{6 + 9 + 15}{2} = 15$

So,

$s - a = 15 - 6 = 9 , s - b = 15 - 9 = 6 , s - c = 15 - 15 = 0$

$\text{Using"color(blue)" Heron's formula :}$ $\text{area of the triangle is :}$

$\Delta = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

$\implies \Delta = \sqrt{15 \left(9\right) \left(6\right) \left(0\right)} = 0$

So the given length of sides can not forms triangle .