Find the area of each triangle ABC.

a = 76.3 ft, b = 109 ft, c = 98.8 ft

Verified step by step guidance

Identify the given sides of the triangle: \(a = 76.3\) ft, \(b = 109\) ft, and \(c = 98.8\) ft. Since all three sides are known, we can use Heron's formula to find the area.

Calculate the semi-perimeter \(s\) of the triangle using the formula: \(s = \frac{a + b + c}{2}\). This value is half the perimeter of the triangle.

Apply Heron's formula for the area \(A\) of the triangle: \(A = \sqrt{s(s - a)(s - b)(s - c)}\). This formula uses the semi-perimeter and the lengths of the sides.

Substitute the values of \(a\), \(b\), \(c\), and \(s\) into the formula to set up the expression for the area.

Evaluate the expression under the square root first, then take the square root to find the area of the triangle in square feet.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Triangle Area Using Heron's Formula

Heron's formula calculates the area of a triangle when all three side lengths are known. It uses the semi-perimeter, s = (a + b + c) / 2, and the area is found by √[s(s - a)(s - b)(s - c)]. This method is useful when no height or angles are given.

Semi-perimeter of a Triangle

The semi-perimeter is half the perimeter of the triangle, calculated as s = (a + b + c) / 2. It is a key intermediate value in Heron's formula and helps simplify the area calculation by relating the sides to the area.

Properties of Triangle Side Lengths

Understanding the triangle inequality theorem is essential to verify if the given sides can form a valid triangle. The sum of any two sides must be greater than the third side, ensuring the triangle exists before applying area formulas.

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