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Triangle RST has sides measuring 22 inches and 13 inches and a perimeter of 50 inches. What is the area of triangle RST? Round to the nearest square inch.

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

A. 19 square inches
B. 37 square inches
C. 60 square inches
D. 95 square inches


Sagot :

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]
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