Problem 23 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=44\) in., \(b=66\) in., \(c=88\) in.

Step-by-Step Solution

Answer

The area of triangle ABC is approximately 4470 square inches.

Step 1: Understand the Problem

We are given the side lengths of triangle \(ABC\): \(a = 44\) inches, \(b = 66\) inches, and \(c = 88\) inches. We need to find the area of the triangle using these side lengths. For a triangle with three known sides, the area can be calculated using Heron's formula.

Step 2: Calculate the Semi-Perimeter

Heron's formula requires us to first find the semi-perimeter \(s\) of the triangle, which is half the sum of the sides. Calculate \(s\) as follows:\[s = \frac{a + b + c}{2} = \frac{44 + 66 + 88}{2} = 99 \text{ inches}.\]

Step 3: Apply Heron's Formula

Heron's formula for the area \(A\) of a triangle is:\[A = \sqrt{s(s-a)(s-b)(s-c)}\]Substitute the values for \(s\), \(a\), \(b\), and \(c\):\[A = \sqrt{99 \times (99-44) \times (99-66) \times (99-88)}\]This simplifies to:\[A = \sqrt{99 \times 55 \times 33 \times 11}\]

Step 4: Calculate the Area

Now calculate the numerical value:\[A \approx \sqrt{19958475} \approx 4470.79 \text{ square inches}.\] Round the area to three significant digits:\[A \approx 4470 \text{ square inches}.\]

Key Concepts

Understanding Triangle Area

When solving for a triangle's area, especially when all three side lengths are given, we need a technique beyond simple base and height. This is where Heron's formula shines. It allows us to determine the area of a triangle when we know the lengths of its sides.For any triangle with sides labeled as \(a\), \(b\), and \(c\), Heron's formula makes it possible to find the area with ease. This approach is advantageous because it does not require the knowledge of the height, which can be complicated to find for scalene triangles, where all sides are different lengths.To use Heron's formula, the calculation of the semi-perimeter is essential, which we will cover in detail in the next section.

Grasping the Concept of Semi-Perimeter

The semi-perimeter of a triangle is an intermediate step in using Heron's formula for the area calculation.Here's the process broken down:

• First, you sum up all the sides of the triangle: \(a\), \(b\), and \(c\).
• Then, divide this sum by 2 to get the semi-perimeter \(s\).

For example, if you have a triangle with side lengths of 44 inches, 66 inches, and 88 inches, you sum them up to get 198 inches.Dividing by 2 gives you the semi-perimeter:\[s = \frac{44 + 66 + 88}{2} = 99 \text{ inches.}\]This semi-perimeter is crucial as it's used in Heron's formula to find the area, making it a key building block in the process.

Approaching Mathematical Problem Solving

Successful mathematical problem solving involves breaking down complex problems into manageable steps. Let's look at how this applies to finding a triangle's area through Heron's formula.The approach is as follows:

• Analyze the Problem: Understand what is given and what needs to be found. In this case, the side lengths of the triangle and the area, respectively.
• Organize the Information: List the known values: side lengths \(a\), \(b\), and \(c\).
• Calculate Incrementally: Start by determining the semi-perimeter, an essential step preceding the application of Heron's formula.
• Apply the Formula: Once components like the semi-perimeter are calculated, substitute these values into Heron's formula.
• Solve Carefully: Work through the arithmetic to find the area, ensuring to check calculations to avoid errors.

This step-by-step methodical approach is instrumental in tackling mathematical problems effectively, fostering precision and confidence in solving geometry problems.