How do you use Heron's formula to find the area of a triangle with sides of lengths #8 #, #3 #, and #9 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #8   #, #3    #, and #9    #?

Sadie Adams · Accepted Answer

#A~~11.8"# square units rounded to one decimal place.

Heron's formula for the area of a triangle is #A=sqrt(s(s-a)(s-b)(s-c))#, where #s# is the semiperimeter, which is half the perimeter. #s=(a+b+c)/2#, where #a=8, b=3, and c=9#. #s=(8+3+9)/2# Simplify. #s=20/2# #s=10# Heron's Formula #A=sqrt(s(s-a)(s-b)(s-c))# Substitute the known values into the equation and solve. #A=sqrt(10(10-8)(10-3)(10-9)# Simplify. #A=sqrt((10)(2)(7)(1))# #A=sqrt140# #A~~11.8"# square units rounded to one decimal place.

Aaliyah Anderson · Answer

undefined To use Heron's formula to find the area of a triangle with side lengths of 8, 3, and 9, follow these steps:

1. Calculate the semiperimeter of the triangle, denoted as $ s $, by adding the lengths of all three sides and dividing by 2.
2. Use the semiperimeter to find the area using Heron's formula, which states that the area ($ A $) of a triangle with side lengths $ a $, $ b $, and $ c $ is given by: 
$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

Now, let's calculate:

1. Semiperimeter $ s $:
$$ s = \frac{8 + 3 + 9}{2} = \frac{20}{2} = 10 $$

2. Use Heron's formula:
$$ A = \sqrt{10(10 - 8)(10 - 3)(10 - 9)} $$

$$ A = \sqrt{10(2)(7)(1)} $$

$$ A = \sqrt{140} $$

$$ A \approx 11.83 $$

So, the area of the triangle with side lengths 8, 3, and 9 is approximately 11.83 square units.