Certainly! Let's solve this step-by-step.
1. Understand the given information:
- The sides of the triangle are [tex]\( a = 6 \)[/tex] inches, [tex]\( b = 8 \)[/tex] inches, and [tex]\( c = 6 \)[/tex] inches.
- The perimeter of the triangle is 20 inches.
2. Calculate the semi-perimeter:
The semi-perimeter [tex]\( s \)[/tex] is given by:
[tex]\[
s = \frac{\text{Perimeter}}{2}
\][/tex]
Substituting the given perimeter:
[tex]\[
s = \frac{20}{2} = 10 \text{ inches}
\][/tex]
3. Apply Heron's formula:
Heron's formula for the area [tex]\( A \)[/tex] of a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] and semi-perimeter [tex]\( s \)[/tex] is:
[tex]\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\][/tex]
4. Substitute the sides and semi-perimeter into Heron's formula:
[tex]\[
A = \sqrt{10 \times (10 - 6) \times (10 - 8) \times (10 - 6)}
\][/tex]
5. Simplify the terms inside the square root:
[tex]\[
A = \sqrt{10 \times 4 \times 2 \times 4}
\][/tex]
[tex]\[
A = \sqrt{10 \times 4 \times 8}
\][/tex]
[tex]\[
A = \sqrt{10 \times 32}
\][/tex]
[tex]\[
A = \sqrt{320}
\][/tex]
6. Calculate the square root:
[tex]\[
A \approx 17.89 \text{ square inches}
\][/tex]
7. Compare with the given options:
The options are 15 square inches, 76 square inches, 186 square inches, and 215 square inches. The area we calculated is approximately 17.89 square inches, which is closest to 15 square inches.
Therefore, approximately 15 square inches of fabric were used to make the triangular flag.
