Question

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

Answer 1

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

The area of triangle by using Herio's formula

`\Delta=\sqrt{s(s-a)(s-b)(s-c)}`

`=\sqrt{15(15-6)(15-9)(15-15)}`

`=0`

The area of the triangle will be zero

Answer 2

Triangle Inequality Theorem :

`"The sum of the lengths of any two sides of a triangle"`

`"must be greater than the length of the third side."`

Explanation:

Let ,

`a=6 ,b=9 and c=15`

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

So, according to Triangle Inequality Theorem ,

`""color(red)"it is not possible to construct a triangle."`

OR

`:." Semi perimeter of triangle is :"`

`s=(a+b+c)/2=(6+9+15)/2=15`

So,

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

`"Using"color(blue)" Heron's formula :"` `"area of the triangle is :"`

`Delta=sqrt(s(s-a)(s-b)(s-c))`

`=>Delta=sqrt(15(9)(6)(0))=0`

So the given length of sides can not forms triangle .