Polygon ABCD: Identify Correct Geometric Statements

Hey guys! Let's dive into some geometry and figure out which statements are true about this polygon. We've got the coordinates for a polygon ABCD, and we need to check out some facts about its sides and area. This is going to be a fun ride, so buckle up!

Understanding the Polygon ABCD

To kick things off, we're given the coordinates of the vertices of our polygon: A(-2, 2), B(1, 5), C(2, 3), and D(-1, 0). These coordinates are our starting point for figuring out the lengths of the sides and the area of the polygon. Geometry problems like these are super cool because they combine algebra and visual thinking. We can use the distance formula to find the lengths of the sides and break down the polygon into simpler shapes, like triangles, to calculate the area. So, let's get our hands dirty and start crunching some numbers!

When we consider the polygon, it's essential to have a clear picture of what we're dealing with. Think of these points plotted on a graph, creating a four-sided shape. Visualizing it can help us anticipate the lengths and the overall shape. Are we looking at a square, a rectangle, or something more irregular? Answering these questions in your mind can guide you to the next steps effectively. Now, let's roll up our sleeves and get into the specifics of each statement. We'll use the distance formula and some area calculation techniques to verify which statements hold true for our polygon ABCD.

Moreover, having a strong grasp of coordinate geometry is extremely beneficial here. It's like having a superpower that allows us to translate geometric figures into algebraic equations and vice versa. For instance, we can find the distance between two points using the distance formula, which is derived from the Pythagorean theorem. This powerful connection between geometry and algebra is the backbone of many geometric calculations, and understanding it thoroughly can make problems like this a piece of cake. So, keep your formulas handy, and let's embark on this geometric adventure together! We're going to dissect this polygon and reveal its secrets, one step at a time.

Verifying the Side Lengths

Statement A: BC = √5 units

Let's start by checking if the length of side BC is indeed √5 units. To do this, we'll use the distance formula, which is given by:

√[(x₂ - x₁)² + (y₂ - y₁)²]

Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. For points B(1, 5) and C(2, 3), we plug in the coordinates:

BC = √[(2 - 1)² + (3 - 5)²] = √[(1)² + (-2)²] = √(1 + 4) = √5

So, statement A is correct! The length of BC is indeed √5 units. Yay, we got one right! This first victory boosts our confidence and sets the stage for tackling the remaining statements. But remember, in math, showing your work is just as important as getting the correct answer. It helps you understand the process and ensures that you can apply the same techniques to other problems. Plus, it's super satisfying to see how the formula transforms numbers into geometric truths. So, with this win under our belt, let's keep our momentum going and jump into verifying statement B. We're on a roll, and there's no stopping us now!

Statement B: AB = √18 units

Now, let's move on to statement B and find the length of side AB. Again, we'll use the distance formula. This time, our points are A(-2, 2) and B(1, 5). Plugging these coordinates into the formula gives us:

AB = √[(1 - (-2))² + (5 - 2)²] = √[(1 + 2)² + (3)²] = √[(3)² + (3)²] = √(9 + 9) = √18

Statement B is also correct! The length of AB is √18 units. Awesome! We're batting a thousand here, and it feels great to confirm another statement. But let's not get too comfortable; we still have more to explore. Each correct verification is like adding a piece to the puzzle, bringing us closer to the complete picture of our polygon. So far, we've seen the magic of the distance formula in action, and we've experienced the satisfaction of seeing our calculations match the given statements. Now, with two down and more to go, let's keep the excitement alive and dive into statement C. We're on a mission to uncover the truths of this polygon, and we won't stop until we've checked every single statement!

Statement C: CD = √22 units

Alright, let's tackle statement C and see if the length of side CD is √22 units. You guessed it – we're dusting off our trusty distance formula once again. This time, we're working with points C(2, 3) and D(-1, 0). Let's plug those coordinates in:

CD = √[(-1 - 2)² + (0 - 3)²] = √[(-3)² + (-3)²] = √(9 + 9) = √18

Wait a minute! We found that CD = √18, but statement C says CD = √22. Statement C is incorrect! Woohoo, we caught an error! It's not just about finding the correct statements; identifying the wrong ones is just as crucial. This is where we need to be extra careful and double-check our calculations to make sure we haven't made any mistakes. In this case, our calculations clearly show that the length of CD is √18, not √22. This little victory reminds us that attention to detail is key in math. So, let's celebrate our detective work and then get ready to move on to the final statement about the area. We're on the home stretch now, and we're bringing our A-game to ensure we nail the rest of this problem!

Calculating the Area

Statement D: Area of ABCD ≈ 9.49 units²

Now, let's get our hands dirty with the area calculation. This is where things get a little more interesting. To find the area of polygon ABCD, we can divide it into two triangles. A common approach is to split it along the diagonal AC, creating triangles ABC and ADC. Then, we can find the area of each triangle and add them up to get the total area of the polygon.

The formula for the area of a triangle given the coordinates of its vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

First, let's calculate the area of triangle ABC using the coordinates A(-2, 2), B(1, 5), and C(2, 3):

Area_ABC = (1/2) |(-2)(5 - 3) + (1)(3 - 2) + (2)(2 - 5)| = (1/2) |(-2)(2) + (1)(1) + (2)(-3)| = (1/2) |-4 + 1 - 6| = (1/2) |-9| = 4.5

Next, we'll find the area of triangle ADC using the coordinates A(-2, 2), D(-1, 0), and C(2, 3):

Area_ADC = (1/2) |(-2)(0 - 3) + (-1)(3 - 2) + (2)(2 - 0)| = (1/2) |(-2)(-3) + (-1)(1) + (2)(2)| = (1/2) |6 - 1 + 4| = (1/2) |9| = 4.5

Now, we add the areas of the two triangles to get the area of polygon ABCD:

Area_ABCD = Area_ABC + Area_ADC = 4.5 + 4.5 = 9

So, the area of polygon ABCD is 9 square units. Statement D says the area is approximately 9.49 square units, which is pretty close but not exactly correct. Statement D is incorrect!

Wow, we've made it through all the statements! Calculating the area involved a bit more work, but we broke it down step by step and conquered it. It's amazing how we can use simple formulas to uncover so much about shapes and figures. And remember, even if our final answer doesn't perfectly match the options, being close and understanding the process is a huge win. So, let's take a moment to appreciate how far we've come and celebrate our geometric prowess! We've shown that we're not just calculators; we're problem-solving superstars!

Conclusion: Correct Statements

Okay, guys, after carefully analyzing each statement, we've determined that:

• Statement A: BC = √5 units (Correct)
• Statement B: AB = √18 units (Correct)
• Statement C: CD = √22 units (Incorrect)
• Statement D: Area of ABCD ≈ 9.49 units² (Incorrect)

Therefore, the three correct statements are A and B. We nailed it! High fives all around! We started with a polygon defined by coordinates and ended up unraveling its secrets. We calculated side lengths using the distance formula, split the polygon into triangles, and computed areas. This is what math is all about – taking a problem, breaking it down, and conquering it with our skills and knowledge. So, let's carry this confidence and problem-solving spirit with us as we tackle new challenges. We've proven that we're not just good at math; we're awesome at it!

I hope you enjoyed this geometric adventure as much as I did. Remember, every problem is a chance to learn something new and flex our mathematical muscles. Keep practicing, keep exploring, and keep shining! You guys rock! And who knows, maybe we'll meet again to tackle another exciting math puzzle. Until then, keep those calculations sharp and those problem-solving skills on point. Adios!