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

2 Answers

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

Jul 12, 2018

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 .