How do you use Heron's formula to find the area of a triangle with sides of lengths #7 #, #3 #, and #9 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #7   #, #3    #, and #9    #?

Evelyn Arboleda · Accepted Answer

#Area=8.7856# square units

Heron's formula for finding area of the triangle is given by #Area=sqrt(s(s-a)(s-b)(s-c))# Where #s# is the semi perimeter and is defined as #s=(a+b+c)/2# and #a, b, c# are the lengths of the three sides of the triangle. Here let #a=7, b=3# and #c=9# #implies s=(7+3+9)/2=19/2=9.5# #implies s=9.5# #implies s-a=9.5-7=2.5, s-b=9.5-3=6.5 and s-c=9.5-9=0.5# #implies s-a=2.5, s-b=6.5 and s-c=0.5# #implies Area=sqrt(9.5*2.5*6.5*0.5)=sqrt77.1875=8.7856# square units #implies Area=8.7856# square units

Ella Armstrong · Answer

undefined To find the area of a triangle using Heron's formula with side lengths of 7, 3, and 9, first calculate the semi-perimeter of the triangle, denoted as $ s $, which is the sum of the lengths of the sides divided by 2. Then, apply Heron's formula, which states that the area ($ A $) of a triangle with side lengths $ a $, $ b $, and $ c $ and semi-perimeter $ s $ is given by:

$$ A = \sqrt{s(s-a)(s-b)(s-c)} $$

Here's the step-by-step calculation:

1. Calculate the semi-perimeter ($ s $):

$$ s = \frac{7 + 3 + 9}{2} = \frac{19}{2} $$

2. Apply Heron's formula:

$$ A = \sqrt{\frac{19}{2} \left(\frac{19}{2}-7\right) \left(\frac{19}{2}-3\right) \left(\frac{19}{2}-9\right)} $$

3. Simplify and compute the square root:

$$ A = \sqrt{\frac{19}{2} \times \frac{5}{2} \times \frac{11}{2} \times \frac{1}{2}} $$

$$ A = \sqrt{\frac{19 \times 5 \times 11 \times 1}{2 \times 2 \times 2 \times 2}} $$

$$ A = \sqrt{\frac{1045}{8}} $$

4. Evaluate the square root:

$$ A = \frac{\sqrt{1045}}{\sqrt{8}} $$

$$ A \approx \frac{32.31}{2.83} $$

$$ A \approx 11.42 $$

So, the area of the triangle with side lengths 7, 3, and 9 is approximately 11.42 square units.