Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Find the area of each triangle. \(a=25.4\) yards, \(b=38.2\) yards, \(c=19.8\) yards

Step-by-Step Solution

Answer

The area of the triangle is approximately 222.64 square yards.

Step 1: Understanding Heron's Formula

To find the area of a triangle with sides of length \(a\), \(b\), and \(c\), we can use Heron's formula. Heron's formula states: \[A = \sqrt{s(s-a)(s-b)(s-c)}\]where \(s\) is the semi-perimeter of the triangle calculated as \(s = \frac{a+b+c}{2}\).

Step 2: Calculate the Semi-Perimeter

Using the formula for the semi-perimeter, calculate:\[s = \frac{a+b+c}{2} = \frac{25.4 + 38.2 + 19.8}{2}\]This results in:\[s = \frac{83.4}{2} = 41.7 \, \text{yards}\]

Step 3: Apply Heron's Formula to Find Area

Now that we have the semi-perimeter \(s\), apply Heron's formula:\[A = \sqrt{41.7(41.7 - 25.4)(41.7 - 38.2)(41.7 - 19.8)}\]Calculate each component:\[41.7 - 25.4 = 16.3 \]\[41.7 - 38.2 = 3.5 \]\[41.7 - 19.8 = 21.9 \]Substitute these values back into the formula:\[A = \sqrt{41.7 \times 16.3 \times 3.5 \times 21.9}\]

Step 4: Calculate the Area

Calculate the product under the square root:\[A = \sqrt{41.7 \times 16.3 \times 3.5 \times 21.9} = \sqrt{49623.5545}\]Finally, find the square root to get the area:\[A \approx 222.64 \, \text{square yards}\]

Key Concepts

Triangle Area

A triangle is a fundamental shape in geometry, consisting of three sides and three corners or angles. Calculating the area of a triangle is a common task in both geometry and trigonometry. There are different methods to find the area, depending on the available information about the triangle. Heron's formula is a versatile method that allows finding the area when you know the lengths of all three sides. This formula involves calculating the semi-perimeter and then using it to find the area through a specific square root formula. By understanding these steps, you gain insight into the properties and measurements of a triangle.

Geometry

Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. Triangles play a vital role in geometry, serving as simple yet powerful shapes for examining various geometric principles. Understanding the geometry of a triangle includes knowing about its angles, sides, and the relationships between these elements. Triangles can be classified by their side length as equilateral, isosceles, or scalene, and by their angle type as acute, right, or obtuse. Each type has unique properties that help in solving geometric problems, including finding areas and understanding spatial relationships.

Trigonometry

Trigonometry is a branch of mathematics focusing on the relationships between the angles and sides of triangles. Although Heron's formula does not directly use trigonometric functions, understanding trigonometry can enhance insights into triangles. For instance, basic trigonometric concepts include functions like sine, cosine, and tangent, which relate angles to side lengths. These functions are essential for solving many practical problems, like those in physics or engineering, where triangles frequently appear. In relation to Heron's formula, trigonometry aids in understanding how the side lengths and angles interact to form the triangle's shape and size.

Semi-Perimeter

The semi-perimeter of a triangle is a helpful intermediate value used in Heron's formula. It is calculated as half of the triangle's perimeter, or the sum of its sides. The formula is given by \(s = \frac{a+b+c}{2}\), where \(a\), \(b\), and \(c\) are the lengths of the triangle's sides. The concept of semi-perimeter simplifies the calculation of the area by breaking the process down into manageable steps. By calculating \(s\), each term in Heron's formula can be simplified, eventually leading to the determination of the triangle's area. Understanding the semi-perimeter is key to efficiently utilizing Heron's method to solve triangle-related problems.