To determine the area of Lila's triangular flag using the given perimeter and Heron's formula, follow these steps:
1. Understand the problem:
- We are given the perimeter of the triangular flag, which is 20 inches.
- We need to calculate the area of the triangle and then choose the closest value from the given options.
2. Assume the triangle is equilateral:
- An equilateral triangle is one where all sides are equal.
- Given the perimeter is 20 inches, if we divide this equally among the three sides, each side will be:
[tex]\[
\frac{20}{3} \approx 6.67 \text{ inches}
\][/tex]
- Let the sides of the triangle be [tex]\(a = b = c = 6.67\)[/tex] inches.
3. Calculate the semi-perimeter:
- The semi-perimeter [tex]\(s\)[/tex] is half of the perimeter:
[tex]\[
s = \frac{20}{2} = 10 \text{ inches}
\][/tex]
4. Use Heron's formula:
- Heron's formula is given by:
[tex]\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\][/tex]
- Substituting in the values:
[tex]\[
\text{Area} = \sqrt{10 \times (10 - 6.67) \times (10 - 6.67) \times (10 - 6.67)}
\][/tex]
- This simplifies to:
[tex]\[
\text{Area} = \sqrt{10 \times 3.33 \times 3.33 \times 3.33}
\][/tex]
5. Calculate the area:
- After carrying out the calculations (which we determined previously):
[tex]\[
\text{Area} \approx 19.245 \text{ square inches}
\][/tex]
6. Choose the closest value:
- Among the given options, the one closest to [tex]\(19.245\)[/tex] square inches is [tex]\(15\)[/tex] square inches.
Thus, the approximate area of fabric used to make the triangular flag is 15 square inches.
