Find the area of a triangular park ABC with sides 50m, 80m, and 120m using Heron's formula.
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Step-by-Step Solution
Step 1: Calculate the semi-perimeter
Heron's formula requires the semi-perimeter, denoted by s. This is half the sum of the lengths of the three sides of the triangle. Given the side lengths a=50 m, b=80 m, and c=120 m, we sum them up and divide by 2.
Step 2: Substitute values into semi-perimeter formula
Substituting the given side lengths into the formula, we get s=250+80+120=2250=125 m.
Step 3: Apply Heron's formula
Heron's formula provides a way to calculate the area of a triangle when only the lengths of its three sides are known. The formula is Area=s(s−a)(s−b)(s−c), where s is the semi-perimeter and a, b, c are the side lengths.
Step 4: Substitute values into Heron's formula
Now we substitute the calculated semi-perimeter s=125 m and the given side lengths a=50 m, b=80 m, c=120 m into Heron's formula.
Step 5: Calculate the area
Performing the subtractions inside the parentheses and then multiplying the terms, we get Area=125×75×45×5=2109375. Finally, we calculate the square root to find the area, which is approximately 1452.36 m2.