To determine the area of triangle PQR, we can use Heron's formula, which computes the area of a triangle when the lengths of all three sides are known.
1. Identify the given values:
- [tex]\( a = 9 \)[/tex] feet
- [tex]\( b = 10 \)[/tex] feet
- Perimeter [tex]\( P = 24 \)[/tex] feet
2. Determine the length of the third side, [tex]\( c \)[/tex]:
- Since the perimeter of the triangle is the sum of all its sides, we have:
[tex]\[
a + b + c = P
\][/tex]
Substitute the known values:
[tex]\[
9 + 10 + c = 24
\][/tex]
Simplify to find [tex]\( c \)[/tex]:
[tex]\[
c = 24 - 9 - 10
\][/tex]
[tex]\[
c = 5 \) feet
\][/tex]
3. Calculate the semi-perimeter, [tex]\( s \)[/tex], of the triangle:
- The semi-perimeter is half of the perimeter, given by:
[tex]\[
s = \frac{P}{2}
\][/tex]
Substitute the value of [tex]\( P \)[/tex]:
[tex]\[
s = \frac{24}{2}
\][/tex]
[tex]\[
s = 12 \) feet
\][/tex]
4. Apply Heron’s formula to find the area [tex]\( A \)[/tex]:
- Heron's formula is given by:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substitute [tex]\( s \)[/tex], [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[
A = \sqrt{12(12 - 9)(12 - 10)(12 - 5)}
\][/tex]
Simplify inside the square root:
[tex]\[
A = \sqrt{12 \times 3 \times 2 \times 7}
\][/tex]
Continue simplifying:
[tex]\[
A = \sqrt{12 \times 3 \times 2 \times 7}
\][/tex]
[tex]\[
A = \sqrt{504}
\][/tex]
5. Calculate the square root:
- The square root of 504 is approximately:
[tex]\[
A \approx 22.44994432064365
\][/tex]
6. Round to the nearest square foot:
- The nearest square foot is:
[tex]\[
A \approx 22
\][/tex]
Therefore, the area of triangle PQR, rounded to the nearest square foot, is 22 square feet. Thus, the correct choice is 22 square feet.
