Problem 33 - Precalculus with Limits

Question

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 8\), \(b = 12\), \(c = 17\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 38.49 square units.

Step 1: Calculate semi-perimeter of the triangle

First, you need to calculate the semi-perimeter of the triangle. This is given by the formula \( s = (a + b + c) / 2 \). So, \( s = (8 + 12 + 17) / 2 = 18.5 \).

Step 2: Apply Heron's Formula

Next, plug the values into Heron's formula to find the area of the triangle. The formula is \( A = \sqrt{s(s - a)(s - b)(s - c)} \). So, substituted with the values we have \( A = \sqrt{18.5(18.5 - 8)(18.5 - 12)(18.5 - 17)} \).

Step 3: Solve the expression

Finally, solve the above expression to get the area of the triangle. Calculation will give \( A = \sqrt{18.5 \cdot 10.5 \cdot 6.5 \cdot 1.5} \approx 38.4852813742 \)

Key Concepts

Triangle Area Calculation

When it comes to finding the area of a triangle, there are several methods, each tailored for different types of triangles and available information. However, for cases where you only know the lengths of the three sides, a method known as Heron's Formula becomes invaluable. This mathematical tool enables you to calculate the area of a triangle without needing to know the height or any angle measures. Heron's formula states that if a triangle has sides of lengths a, b, and c, then its area (A) can be found using the following expression:

\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
where s represents the semi-perimeter of the triangle. This formula works for all types of triangles, including scalene, where no two sides are equal, ensuring broad applicability. It's especially useful in situations such as land surveying, where direct measurements of height might not be feasible.

Semi-perimeter of a Triangle

The semi-perimeter of a triangle, often represented by the letter s, is essentially half of the triangle's perimeter. It is a critical part of Heron's Formula, acting as a stepping-stone to calculate the area. To find the semi-perimeter, you add up the lengths of the three sides of the triangle, then divide the sum by two. So, if a triangle has side lengths a, b, and c, the semi-perimeter is given by:

\[ s = \frac{a + b + c}{2} \]
The semi-perimeter is not only important for calculating area with Heron's formula but is also used in other aspects of geometry and trigonometry, making it a universally useful concept.

Mathematical Problem-solving

Mathematical problem-solving is a structured approach to addressing challenges that require numerical solutions. It typically involves several steps, starting with understanding the problem, which might be finding a triangle's area. Then, you devise a strategy, such as deciding to use Heron's Formula because only side lengths are known. Next, carry out the strategy by applying the formula and performing necessary calculations, as we did by substituting known lengths into the formula to find the semi-perimeter and subsequently the area.

If faced with any complications, it's essential to re-evaluate and adjust the strategy, ensuring that each step has been correctly executed. To conclude, validating the final answer is paramount. The process helps build strong analytical and cognitive skills, particularly in understanding how mathematical principles apply to real-world situations.