Problem 40 - Precalculus with Limits

Question

Using Heron's Area Formula use Heron's Area Formula to find the area of the triangle. $$ a=75.4, \quad b=52, \quad c=52 $$

Step-by-Step Solution

Answer

The area of the triangle is 693.9 square units

Step 1: Calculate Semi-Perimeter

The semi-perimeter (s) of the triangle can be calculated using the formula: \(s = \frac{a + b + c}{2}\). Substituting the given values: \(s = \frac{75.4 + 52 + 52}{2} = 89.7\)

Step 2: Apply Heron's Formula

To find the area (A) of the triangle, plug the calculated semi-perimeter and the given side lengths into Heron's formula: \(A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{89.7(89.7 - 75.4)(89.7 - 52)(89.7 - 52)} = \sqrt{89.7(14.3)(37.7)(37.7)}\)

Step 3: Calculate the Area

Perform the multiplication and take the square root to get the area: \(A = \sqrt{481485.7673} = 693.9\) units squared

Key Concepts

Semi-Perimeter

In geometry, the semi-perimeter of a triangle is an essential concept when it comes to calculating the area using Heron's Formula. The semi-perimeter is half of the triangle's perimeter.
Simply put, it’s the average of the sum of the three sides of the triangle.
To find the semi-perimeter \(s\) for a triangle with sides \(a, b,\) and \(c\), the formula is:

• \(s = \frac{a + b + c}{2}\)

This value plays a pivotal role in Heron's Formula, simplifying the computation of the area by reducing it to just one convenient expression.
For instance, with side lengths \(a = 75.4, b = 52,\) and \(c = 52\), you can calculate the semi-perimeter as follows:

• \(s = \frac{75.4 + 52 + 52}{2} = 89.7\)

Triangle Area Calculation

Calculating the area of a triangle can be done using various methods, but Heron's Formula is a robust way to handle triangles when only the lengths of the sides are known.
After obtaining the semi-perimeter, Heron’s formula makes it possible to compute the area \(A\) efficiently.
Here's the formula:

• \(A = \sqrt{s(s - a)(s - b)(s - c)}\)

This formula requires substituting the semi-perimeter and the side lengths of the triangle.
Let's consider our specific example:

• \(A = \sqrt{89.7(89.7 - 75.4)(89.7 - 52)(89.7 - 52)}\)

Breaking down each step ensures you understand how each component contributes to the final area calculation.

Geometry

Geometry involves understanding shapes, sizes, and the relative position of figures as well as the properties of space.
One of the intriguing topics within geometry is the study of triangles, their properties, and various formulas to calculate their dimensions.
Heron's Formula is a more advanced concept within this field, providing a reliable method for finding the area of a triangle when the lengths of all three sides are known.
This formula underscores the broader geometric principle of transforming and manipulating different properties to arrive at a desired result.

• By using the semi-perimeter and manipulating it through subtraction and multiplication, you calculate an area that reflects the spatial dimensions of the triangle accurately.

Thus, the elegance in Heron’s approach lies in its general applicability, making it a fundamental tool in both theoretical and applied geometry.