In Exercises 13–16, find the area of the triangle having the given measurements. Round to the nearest square unit. a = 2 meters, b = 4 meters, c = 5 meters

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Identify the formula for the area of a triangle using Heron's formula: \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s \) is the semi-perimeter of the triangle.

Calculate the semi-perimeter \( s \) using the formula \( s = \frac{a+b+c}{2} \). Substitute the given side lengths: \( a = 2 \), \( b = 4 \), and \( c = 5 \).

Substitute the values of \( s \), \( a \), \( b \), and \( c \) into Heron's formula: \( A = \sqrt{s(s-a)(s-b)(s-c)} \).

Simplify the expression inside the square root by performing the arithmetic operations: \( s-a \), \( s-b \), and \( s-c \).

Calculate the square root of the resulting value to find the area of the triangle.

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Triangle Area Formulas

The area of a triangle can be calculated using various formulas, with one of the most common being Heron's formula. This formula is particularly useful when the lengths of all three sides are known. It states that the area can be found using the semi-perimeter (s) and the side lengths (a, b, c) as follows: Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.

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Calculating Area of ASA Triangles

Heron's Formula

Heron's formula allows for the calculation of a triangle's area when the lengths of all three sides are known. By first calculating the semi-perimeter, which is half the sum of the side lengths, one can then apply the formula to find the area. This method is particularly advantageous when height is not readily available, making it a versatile tool in trigonometry.

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Quadratic Formula

Properties of Triangles

Understanding the properties of triangles, including the triangle inequality theorem, is essential for determining the feasibility of a triangle with given side lengths. The theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. This ensures that the specified dimensions can indeed form a valid triangle, which is a prerequisite for calculating its area.

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