Lila made a triangular flag to cheer on her favorite sports team. The perimeter of the flag is 20 inches.

Approximately how many square inches of fabric were used to make the triangular flag?

A. 15 square inches
B. 76 square inches
C. 186 square inches
D. 215 square inches

Use Heron's formula: [tex]\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)[/tex]



Answer :

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.