In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit. a = 14 meters, b = 12 meters, c = 4 meters

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: $s=\frac{14+12+4}{2}$.
3. Simplify the expression to find the value of $s$.
4. Use Heron's formula to find the area: $A=\sqrt{s(s-a)(s-b)(s-c)}$.
5. Substitute the values of $s$, $a$, $b$, and $c$ into Heron's formula and simplify to find the area.

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 when the height of the triangle is not known.

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 third side. This theorem is essential for determining whether a set of three lengths can form a triangle. In the given problem, verifying that a = 14 m, b = 12 m, and c = 4 m satisfy this theorem ensures that the area calculated using Heron's formula is valid.