In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit. a = 4 feet, b = 4 feet, c = 2 feet

Verified step by step guidance

1. Calculate the semi-perimeter of the triangle using the formula: \( s = \frac{a + b + c}{2} \).
2. Substitute the given side lengths into the semi-perimeter formula: \( s = \frac{4 + 4 + 2}{2} \).
3. Use Heron's formula to find the area: \( A = \sqrt{s(s-a)(s-b)(s-c)} \).
4. Substitute the values of \( s \), \( a \), \( b \), and \( c \) into Heron's formula.
5. Simplify the expression under the square root and calculate the area, rounding to the nearest square unit.

Key Concepts

Heron's Formula

Heron's formula is a method for calculating the area of a triangle when the lengths of all three sides are known. It states that the area can be found using the formula A = √(s(s-a)(s-b)(s-c)), where 's' is the semi-perimeter of the triangle, calculated as s = (a + b + c) / 2. This formula is particularly useful for triangles that do not have a height readily available.

Semi-perimeter

The semi-perimeter of a triangle is half of its perimeter and is denoted by 's'. It is calculated by adding the lengths of all three sides and dividing by two: s = (a + b + c) / 2. The semi-perimeter is a crucial component in Heron's formula, as it helps simplify the calculation of the area by providing a reference point for the side lengths.

Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem is essential for verifying whether a set of three side lengths can form a triangle. In the given problem, checking the side lengths (4, 4, and 2 feet) against this theorem ensures that they can indeed form a valid triangle before applying Heron's formula.