To solve for the area of the triangle, we can use Heron's Formula. This formula calculates the area of a triangle when all three sides are known.

Step-by-Step Solution:

1. Label the Triangle
   First, let's draw and label the triangle with sides $a = 5$ feet, $b = 7$ feet, and $c = 8$ feet.

          c = 8 ft
       A___________C
       |          /
       |        /
     a=5ft    /
       |    /
       |  /
       |/
      B
       b=7ft
   

2. Calculate the Semi-Perimeter $s$
   The semi-perimeter $s$ is half of the triangle's perimeter: $s = \frac{a + b + c}{2}$ Substituting the values: $s = \frac{5 + 7 + 8}{2} = \frac{20}{2} = 10 \text{ ft}$

3. Apply Heron's Formula
   Heron's formula for the area $A$ of a triangle is: $A = \sqrt{s(s - a)(s - b)(s - c)}$ Substituting the values of $s$, $a$, $b$, and $c$: $A = \sqrt{10(10 - 5)(10 - 7)(10 - 8)}$ Simplify inside the parentheses: $A = \sqrt{10 \times 5 \times 3 \times 2}$ Continue calculating: $A = \sqrt{10 \times 5 \times 3 \times 2} = \sqrt{300}$ Taking the square root: $A \approx 17.32 \text{ square feet}$

Final Answer:

The area of the triangle is approximately 17.32 square feet.

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Would you like additional details on any of these steps?

Here are five related questions to explore further:

1. How would the area change if side lengths were different?
2. What other methods can we use to find the area of a triangle?
3. How can we use trigonometry to verify the area of this triangle?
4. What is the importance of knowing the triangle's semi-perimeter?
5. How would you calculate the perimeter of the triangle?

Tip: Heron’s formula is most useful when you know all three sides but no angles of a triangle.