Problem 28 - Precalculus with Calculus Previews

Question

Use Heron's formula in Problem 27 to find the area of a triangular garden plot if the lengths of the three sides are \(25,32,\) and \(41 \mathrm{~m}\), respectively.

Step-by-Step Solution

Answer

The area of the triangular garden plot is 126 m².

Step 1: Understand Heron's Formula

Heron's Formula is used to calculate the area of a triangle when the lengths of all three sides are known. It is given by: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]where \(s\) is the semi-perimeter of the triangle, and \(a, b,\) and \(c\) are the lengths of the sides.

Step 2: Calculate the Semi-Perimeter

First, we need to calculate the semi-perimeter \(s\) of the triangle. The semi-perimeter \(s\) is given by: \[ s = \frac{a+b+c}{2} \]Substituting the side lengths (\(a = 25, b = 32, c = 41\)), we find: \[ s = \frac{25+32+41}{2} = 49 \].

Step 3: Apply Heron's Formula

Now that we have the semi-perimeter, we can use Heron's Formula to find the area of the triangle:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substituting the values of \(s = 49, a = 25, b = 32,\) and \(c = 41\), we get:\[ A = \sqrt{49(49-25)(49-32)(49-41)} \].

Step 4: Perform the Calculations

Calculate the expressions inside the square root:- \(49 - 25 = 24\)- \(49 - 32 = 17\)- \(49 - 41 = 8\)Now substitute these back into the formula:\[ A = \sqrt{49 \times 24 \times 17 \times 8} \].Calculate the product: \[ 49 \times 24 \times 17 \times 8 = 15904 \].Calculate the square root: \[ A = \sqrt{15904} = 126 \].

Step 5: Conclude the Calculation

After performing all the calculations, we have determined the area of the triangular garden plot is \(126 \text{ m}^2\).

Key Concepts

Triangular Area Calculation

When faced with the challenge of finding the area of a triangle and you know the lengths of all three sides, Heron’s Formula is your best friend. This formula shines in situations where the traditional base-height method is not feasible. Using Heron's Formula involves:

• Calculating the semi-perimeter of the triangle.
• Applying the formula using all three side lengths and the semi-perimeter.

This formula efficiently provides the triangular area without needing to find an altitude. This is particularly useful for triangles that aren’t right-angled or where it's inconvenient to compute the height. Let's look at how these calculations function in detail.

Semi-Perimeter

The semi-perimeter (\(s\)) is a crucial intermediary step in calculating a triangle's area using Heron’s Formula. It's basically half the perimeter. You can find it using the following steps:

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

For instance, in a triangle with sides of lengths 25 m, 32 m, and 41 m, the sum of the side lengths is 98 meters. Dividing this by 2 gives a semi-perimeter of 49 meters.
The semi-perimeter works as a unique value that links the geometry of the triangle to its two-dimensional space, making further complex calculations straightforward.

Geometry Problem Solving

Approaching geometry problems often requires a methodical breakdown of the problem into manageable steps. Understanding and applying Heron's Formula involves several key problem solving skills:

• Identifying and assigning the lengths of the triangle sides.
• Performing precise arithmetic to find the semi-perimeter.
• Substituting these computed values into the formula to determine the area.

Each step demands attention to detail to ensure calculations are correct, as small errors could lead to significant mistakes in the final result. By mastering these steps, you not only solve the problem at hand but also cultivate a systematic approach to tackling a wide range of geometric challenges.