To find the area of triangle PQR given the side lengths and perimeter, we can use Heron's formula. Here is a detailed, step-by-step solution:
### Step 1: Determine the lengths of all three sides
Given the perimeter of the triangle is 24 feet, and two sides are 9 feet and 10 feet, we can find the third side.
[tex]\[ a = 9 \text{ feet} \][/tex]
[tex]\[ b = 10 \text{ feet} \][/tex]
[tex]\[ c = \text{Perimeter} - a - b = 24 - 9 - 10 = 5 \text{ feet} \][/tex]
### Step 2: Calculate the semi-perimeter
The semi-perimeter (s) is half of the perimeter of the triangle.
[tex]\[ s = \frac{\text{Perimeter}}{2} = \frac{24}{2} = 12 \text{ feet} \][/tex]
### Step 3: Apply Heron's formula
Heron's formula is given by:
[tex]\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
Substitute [tex]\( s = 12 \)[/tex], [tex]\( a = 9 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = 5 \)[/tex]:
[tex]\[ \text{Area} = \sqrt{12(12-9)(12-10)(12-5)} \][/tex]
[tex]\[ \text{Area} = \sqrt{12 \times 3 \times 2 \times 7} \][/tex]
[tex]\[ \text{Area} = \sqrt{12 \times 3 \times 2 \times 7} \][/tex]
[tex]\[ \text{Area} = \sqrt{504} \][/tex]
### Step 4: Calculate the numerical value
Calculate the square root of 504:
[tex]\[ \text{Area} \approx 22.44994432064365 \ \text{square feet} \][/tex]
### Step 5: Round to the nearest square foot
Round the area to the nearest whole number:
[tex]\[ \text{Area} \approx 22 \ \text{square feet} \][/tex]
Thus, the area of triangle PQR, rounded to the nearest square foot, is:
[tex]\[ \boxed{22 \ \text{square feet}} \][/tex]
