Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Heron's Area Formula to find the area of the triangle. $$a=2.4, \quad b=2.75, \quad c=2.25$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 2.82 square units.

Step 1: Calculate the Semi-Perimeter

Before the area can be calculated, the semi-perimeter of the triangle must be found. This is done by adding the lengths of the triangle's sides and dividing by 2. The equation for this is \(s = (a + b + c) / 2\).\nUsing the known values \(a = 2.4\), \(b = 2.75\), and \(c = 2.25\), the semi-perimeter is calculated as \(s = (2.4 + 2.75 + 2.25) / 2 = 3.7\).

Step 2: Calculate the Area

Once the semi-perimeter is calculated, the area of the triangle can be found using Heron's Area Formula. This is obtained by substituting the values for \(s\), \(a\), \(b\), and \(c\) into the formula \(A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}\).\nSubstituting the values of \(s = 3.7\), \(a = 2.4\), \(b = 2.75\), and \(c = 2.25\) results in \(A = \sqrt{(3.7 - 2.4) \cdot (3.7 - 2.75) \cdot (3.7 - 2.25) \cdot 3.7}\). Continuing with the calculation, the area \(A\) equals approximately 2.82 square units.

Key Concepts

Understanding the Semi-Perimeter

To find a triangle's area using Heron's Area Formula, you first need to determine the semi-perimeter. This is an essential step because it simplifies the way you will later calculate the area. The semi-perimeter, denoted as \(s\), is half the perimeter of the triangle.
It acts as a central parameter in Heron's formula.The formula to calculate the semi-perimeter is straightforward:

• Add the lengths of all three sides of the triangle.
• Divide the sum by two.

For instance, if a triangle has sides \(a = 2.4\), \(b = 2.75\), and \(c = 2.25\), the semi-perimeter \(s\) is calculated as:\[s = \frac{(2.4 + 2.75 + 2.25)}{2} = 3.7\]Knowing \(s\) makes it easier to use Heron’s formula, as it feeds directly into the multiplication terms needed for the final area calculation.

Triangle Basics

A triangle is a basic geometric shape with three sides, three vertices, and three angles. Triangles can vary greatly in shape and size, but all possess some inherent properties that make area calculations possible.
These include the relationships among the sides and angles, and the concept of a perimeter. For this discussion, it is vital to understand that regardless of the specific type of triangle—be it equilateral, isosceles, or scalene—the properties used in Heron's formula sum up the triangle's whole perimeter initially, which then transitions into a semi-perimeter for further calculations.
The symmetry and form of triangles make them uniquely simple but endlessly useful in geometry and trigonometry.

Area Calculation Using Heron's Formula

Once the semi-perimeter \(s\) of a triangle is known, you can use Heron's formula to calculate the area. This allows you to find the area without needing the height, which is typically required in other area calculations.
The formula is:\[A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}\]Here are the steps to compute the area using Heron's formula:

• Compute the semi-perimeter \(s\).
• Subtract each side length from \(s\), giving you the values \((s - a)\), \((s - b)\), and \((s - c)\).
• Multiply these differences with \(s\) (i.e., \(s \cdot (s-a) \cdot (s-b) \cdot (s-c)\)).
• Take the square root of the result to find the area \(A\).

Using the example values, the area is calculated as approximately 2.82 square units.
This method is efficient and works well for any given triangle's side lengths.