Pentagon Area Calculation: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun geometry problem: calculating the area of a pentagon. Specifically, we'll be tackling a problem from UNIMONTES – MG, where we need to find the area of a pentagon with given vertices. Don't worry; it sounds intimidating, but we'll break it down into easy-to-follow steps. So, let's get started!

Understanding the Problem

The question asks: What is the area of the pentagon with vertices A(0, 0), B(3, 2), C(2, 3), D(1, 5), and E(–2, 1)? The options provided are (A) 14.5, (B) 12.5, (C) 11.5, and (D) 10.5. To solve this, we'll use a method commonly employed in coordinate geometry. This method involves using the determinant of a matrix formed by the coordinates of the vertices.

Why This Method Works

Before we jump into the calculations, let's quickly understand why this method works. The determinant method is essentially an extension of the formula used to find the area of a triangle using coordinates. For a triangle with vertices (x1, y1), (x2, y2), and (x3, y3), the area can be calculated as half the absolute value of the determinant of a matrix formed by these coordinates. This concept extends to polygons with more sides, like our pentagon. By dividing the pentagon into triangles and summing their areas (which are derived from determinants), we can find the total area of the pentagon. Pretty neat, huh?

Breaking Down the Concept for Clarity

Think of it this way: each pair of vertices, when connected to the origin (0,0), forms a triangle. The determinant calculation essentially gives us a signed area of each of these triangles. Some areas might be positive, and some might be negative, depending on the order in which we list the vertices. When we take the absolute value of the sum of these signed areas, and then halve it, we get the total area of the polygon, regardless of its shape or orientation. This method is super useful because it's systematic and works for any polygon, as long as we know the coordinates of its vertices. So, now that we have a solid grasp of the underlying principle, let's roll up our sleeves and get into the actual calculation. Ready to see how it all comes together?

Step-by-Step Calculation

Okay, let's get our hands dirty with the math! To find the area of the pentagon, we'll use the determinant method. This involves setting up a matrix with the coordinates of the vertices and then calculating its determinant.

Setting Up the Matrix

First, we list the coordinates of the vertices in a counter-clockwise direction. This is crucial for getting the correct sign in our determinant calculation. Our vertices are A(0, 0), B(3, 2), C(2, 3), D(1, 5), and E(–2, 1). We'll arrange these in a matrix like this:

| 0  0 |
| 3  2 |
| 2  3 |
| 1  5 |
| -2 1 |
| 0  0 |

Notice that we repeat the first vertex (0, 0) at the end. This is a necessary step in this method to "close the loop," ensuring we correctly account for the area enclosed by the pentagon.

Calculating the Determinant

Next, we calculate the determinant using a method that involves multiplying diagonally and then summing the results. We multiply the numbers diagonally downwards from left to right and add them up, and then we multiply the numbers diagonally upwards from right to left and subtract them. Let's break it down:

• Downward Diagonals: (0 * 2) + (3 * 3) + (2 * 5) + (1 * 1) + (-2 * 0) = 0 + 9 + 10 + 1 + 0 = 20
• Upward Diagonals: (0 * 3) + (2 * 2) + (3 * 1) + (5 * -2) + (1 * 0) = 0 + 4 + 3 - 10 + 0 = -3

Now, we subtract the sum of the upward diagonals from the sum of the downward diagonals: 20 - (-3) = 20 + 3 = 23.

Finding the Area

Finally, to find the area of the pentagon, we take the absolute value of the result and divide it by 2. So, the area is |23| / 2 = 23 / 2 = 11.5. And there we have it! The area of the pentagon is 11.5 square units. This matches option (C) in our list of possible answers. See? It's not so scary when we take it one step at a time.

Verifying the Answer

Now that we've crunched the numbers and arrived at our answer, 11.5, it's always a good idea to take a moment to verify our solution. This is a crucial step in problem-solving, especially in exams, as it helps catch any potential errors. Let's discuss a few ways we can double-check our answer to ensure we're on the right track.

Quick Estimation

One way to get a sense of whether our answer is reasonable is to make a quick estimation. If we were to roughly sketch the pentagon on a coordinate plane, we could try to visualize the area. We might see that the pentagon occupies a space that looks a little larger than a square with sides of length 3. Such a square would have an area of 9, so an area around 11 or 12 seems plausible. This isn't a precise check, but it can help us rule out wildly incorrect answers. For example, if we had calculated an area of 5 or 50, this rough estimation would immediately raise a red flag.

Alternative Methods or Formulas

Another powerful verification technique is to consider if there's an alternative method or formula we could use to calculate the area. In this case, we've used the determinant method, which is quite efficient for polygons defined by their vertices. However, if we were feeling ambitious, we could try dividing the pentagon into triangles and calculating the area of each triangle separately using the standard triangle area formula (1/2 * base * height) or Heron's formula. This would be more time-consuming, but it would provide a completely independent check on our answer. If both methods yield the same result, we can be much more confident in our solution.

Reviewing the Steps

Perhaps the simplest and often most effective way to verify is to meticulously review each step of our calculation. Did we set up the matrix correctly with the vertices in the right order? Did we perform the diagonal multiplications and summations without any arithmetic errors? Did we apply the final area formula correctly, remembering to take the absolute value and divide by 2? Walking back through the process step-by-step can often reveal a small mistake that we might have overlooked initially. It's like proofreading a document – sometimes, reading it slowly and deliberately is the best way to catch errors.

Conclusion

So, there you have it! We've successfully calculated the area of the pentagon using the determinant method and even discussed some ways to verify our answer. The correct answer is indeed (C) 11.5. This problem showcases a cool application of coordinate geometry, and I hope you found this explanation helpful. Remember, the key is to break down complex problems into smaller, manageable steps. Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to explore more geometry problems, feel free to ask. Happy calculating!