Problem 24 - 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. \(a=12\) in., \(b=23\) in., \(c=34\) in.

Step-by-Step Solution

Answer

The area of triangle \(ABC\) is approximately 66.8 square inches.

Step 1: Understand the Problem Using Heron's Formula

To find the area of triangle \(ABC\), we can use Heron's formula since all three sides are known. Heron's formula states that for a triangle with sides \(a, b, c\), the area \(A\) is calculated as \(A = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s\) is the semi-perimeter of the triangle.

Step 2: Calculate the Semi-perimeter

First, calculate the semi-perimeter \(s\) of triangle \(ABC\). The semi-perimeter is given by the formula: \[ s = \frac{a+b+c}{2} \]Substitute the given values:\[ s = \frac{12 + 23 + 34}{2} = \frac{69}{2} = 34.5 \] inches.

Step 3: Apply Heron's Formula to Find the Area

Substitute the semi-perimeter \(s\) and side lengths \(a, b, c\) into Heron's formula:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]\[ A = \sqrt{34.5(34.5-12)(34.5-23)(34.5-34)} \]\[ A = \sqrt{34.5 \times 22.5 \times 11.5 \times 0.5} \]

Step 4: Calculate Specifically Using the Formula

Calculate step-by-step:\[ A = \sqrt{34.5 \times 22.5 \times 11.5 \times 0.5} \]First, perform the multiplications:\[ 34.5 \times 22.5 = 776.25 \]\[ 776.25 \times 11.5 = 8926.875 \]\[ 8926.875 \times 0.5 = 4463.4375 \]Now, take the square root:\[ A = \sqrt{4463.4375} \approx 66.822 \] square inches.

Step 5: Round to Three Significant Digits

Finally, round the area to three significant digits. The area of triangle \(ABC\) is approximately \(66.8\) square inches.

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle can often seem daunting, but with the right tools and understanding, it becomes straightforward. In this exercise, we use **Heron's formula**, which is particularly useful when you know the lengths of all three sides of the triangle. This method eliminates the need for height, making it a versatile solution for various problems. Heron's formula calculates the triangle's area by using half the perimeter, also known as the **semi-perimeter**. It's a powerful shortcut, especially for triangles with varying side lengths. What makes Heron's method so special is its ability to find the area without needing any additional geometric constructs like angles or heights. When you apply this formula, ensure each calculation is done precisely, keeping the side lengths in mind. For this problem, we focus on a triangle with sides 12 in., 23 in., and 34 in. Following the formula closely paves the way for accuracy and ease.

Semi-perimeter

The concept of **semi-perimeter** is crucial when dealing with Heron's formula. It's essentially half of the triangle's perimeter and is calculated using the simple formula:\[s = \frac{a+b+c}{2}\]This step in the equation acts as a starting block, from which we derive further calculations. For example, with side lengths of 12 in., 23 in., and 34 in., the semi-perimeter computes to:\[s = \frac{12 + 23 + 34}{2} = 34.5 \text{ inches}\]The semi-perimeter serves as a foundation for calculating the area, acting as a balance point that pieces together the geometry of the problem neatly. Without calculating the semi-perimeter, applying Heron's formula accurately would not be possible. It's important in this step to double-check your additions and divisions to ensure your calculations are correct.

Significant Digits

When solving mathematical problems, particularly in applied contexts, the concept of **significant digits** is imperative. This concept is employed to ensure precision and clarity in reporting numbers, minimizing errors in interpretation. The significant digits are the digits that carry meaning in expressing the precision of a quantity. When given in problems like the area of triangle calculations, rounding to significant digits ensures numerical accuracy without overcomplicating your answer. In our example, the calculated area of the triangle was initially found to be approximately 66.822 square inches. However, rounding it to three significant digits—as the problem requests—gives a clean, precise result of 66.8 square inches. This process avoids excessive precision, yet maintains the integrity of the answer.

Mathematical Problem Solving

**Mathematical problem solving** is an art that combines systematic thinking with calculated methods. Using formulas like Heron's is just one of many approaches to tackling geometric problems. Understanding each component and their role in the overall solution is key to mastering the problem-solving process. Breaking down the task:

• Start by understanding the problem and identifying known values.
• Select the appropriate formulas to solve the problem.
• Carefully carry out all calculations, checking each step for accuracy.
• Consider the context, such as necessary precision.

Heron's formula is a prime example of how **comprehension** of concepts simplifies mathematical problem solving. Practicing these problems improves confidence and problem-solving efficiency, encouraging students to explore deeper mathematical realms without fearing complexity.