To find the area of Triangle RST with sides measuring 22 inches and 13 inches, and a perimeter of 50 inches, we will follow these steps:
1. Determine the Length of the Third Side:
- The perimeter of Triangle RST is given as 50 inches. We know two sides, 22 inches and 13 inches. The third side (let's call it [tex]\( c \)[/tex]) can be calculated by subtracting the sum of the two known sides from the perimeter:
[tex]\[
c = 50 - 22 - 13 = 15 \text{ inches}
\][/tex]
2. Calculate the Semi-Perimeter:
- The semi-perimeter ([tex]\( s \)[/tex]) is half of the perimeter. Thus:
[tex]\[
s = \frac{50}{2} = 25 \text{ inches}
\][/tex]
3. Apply Heron's Formula:
- Heron's formula for the area of a triangle is:
[tex]\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\][/tex]
- Substitute the known values [tex]\( a = 22 \)[/tex], [tex]\( b = 13 \)[/tex], and [tex]\( c = 15 \)[/tex]:
[tex]\[
\text{Area} = \sqrt{25 \times (25 - 22) \times (25 - 13) \times (25 - 15)}
\][/tex]
- Calculate the intermediate steps:
[tex]\[
s - a = 25 - 22 = 3
\][/tex]
[tex]\[
s - b = 25 - 13 = 12
\][/tex]
[tex]\[
s - c = 25 - 15 = 10
\][/tex]
- Now substitute back into Heron's formula:
[tex]\[
\text{Area} = \sqrt{25 \times 3 \times 12 \times 10}
\][/tex]
- Simplify inside the square root:
[tex]\[
\text{Area} = \sqrt{9000}
\][/tex]
4. Calculate the Square Root:
- The square root of 9000 is approximately:
[tex]\[
\sqrt{9000} \approx 94.86832980505137
\][/tex]
5. Round the Result:
- Round the area to the nearest square inch:
[tex]\[
94.86832980505137 \approx 95
\][/tex]
Therefore, the area of Triangle RST, rounded to the nearest square inch, is 95 square inches. The correct answer is:
[tex]\[
\boxed{95 \text{ square inches}}
\][/tex]
