How do you use Heron's formula to find the area of a triangle with sides of lengths #7 #, #4 #, and #7 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #7   #, #4    #, and #7    #?

Caroline Aucoin · Accepted Answer

#"Area"_triangle = 6sqrt(5)# Heron's formula says that for a triangle with sides of length #a, b, c# and semi-perimeter #s=(a+b+c)/2# #"Area"_triangle = sqrt(s(s-a)(s-b)(s-c))# For the specified triangle #color(white)("XXX")a=7# #color(white)("XXX")b=4# #color(white)("XXX")c=7# #rarrcolor(white)("XXX")s=9# #"Area"_triangle = sqrt(9(2)(5)(2)) = sqrt(3^3xx2^2xx5) = 6sqrt(5)#

Claire Akin · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths $a$, $b$, and $c$, where $s$ is the semi-perimeter of the triangle given by $s = \frac{{a + b + c}}{2}$, and $A$ denotes the area of the triangle, follow these steps:

1. Calculate the semi-perimeter of the triangle using the formula $s = \frac{{a + b + c}}{2}$.
2. Use Heron's formula: $A = \sqrt{s(s - a)(s - b)(s - c)}$.
3. Substitute the values of $a$, $b$, $c$, and $s$ into the formula.
4. Calculate the area using the provided values.

For the triangle with sides of lengths 7, 4, and 7:

1. Calculate the semi-perimeter:
$$s = \frac{{7 + 4 + 7}}{2} = \frac{{18}}{2} = 9$$

2. Apply Heron's formula:
$$A = \sqrt{9(9 - 7)(9 - 4)(9 - 7)}$$

3. Calculate:
$$A = \sqrt{9 \times 2 \times 5 \times 2}$$

$$A = \sqrt{180}$$

4. Simplify:
$$A = \sqrt{36 \times 5}$$

$$A = \sqrt{36} \times \sqrt{5}$$

$$A = 6 \times \sqrt{5}$$

So, the area of the triangle is $6\sqrt{5}$ square units.