Calculating CD Length: Distance Formula Explained
Hey guys! Let's dive into a cool math problem where we'll use the distance formula to find the length of a line segment. We're given a quadrilateral on a coordinate plane, and our mission is to calculate the length of one of its sides, specifically CD. Don't worry, it's not as scary as it sounds! We'll break it down step by step, making sure everyone understands. This is a fundamental concept in coordinate geometry, and understanding it will help you with a bunch of other math problems. So, buckle up, and let's get started. We'll be using the distance formula, which is a key tool for solving problems like these, and by the end, you'll be able to find the length of any line segment on a coordinate plane with confidence.
First things first, what exactly are we dealing with? We have a quadrilateral, which is a fancy word for a four-sided shape. This quadrilateral sits on a coordinate plane, meaning we can pinpoint each of its corners using x and y coordinates. The coordinates of the vertices are given to us: (-4, 1), (-1, 3), (3, 1), and (0, -1). Our goal is to find the length of the line segment CD. But how do we do that? Well, that's where the distance formula comes in handy. It's like a secret weapon for finding the distance between two points on the coordinate plane. And don't worry, it's not super complicated; it's just a formula we can apply to the coordinates of the points to get our answer. Understanding the basics of the coordinate plane is also essential here. The coordinate plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is represented by an ordered pair (x, y), where x is the horizontal distance from the origin (0, 0), and y is the vertical distance. So, when we are given coordinates like (-4, 1), we know that we are dealing with a point located 4 units to the left of the origin and 1 unit up. Cool, right? It's all about understanding how these points are positioned on the plane.
Now, let's talk about the distance formula itself. The distance formula is derived from the Pythagorean theorem, which you might have heard of. It helps us find the distance between two points (x1, y1) and (x2, y2) on a coordinate plane. Here's the formula: Distance = √((x2 - x1)² + (y2 - y1)²). Yeah, it might look a bit intimidating at first, but let's break it down. We're essentially finding the difference in the x-coordinates, squaring it, finding the difference in the y-coordinates, squaring it, adding those two results together, and then taking the square root. That's it! Once you plug in the numbers, it's just basic arithmetic. The square root part might seem tricky, but a calculator will make it easy. So don't let the formula scare you, it is your friend. We are using this formula because the coordinate plane allows us to visualize points and calculate distances using their x and y coordinates. This is especially helpful because we can see the location of our points visually, making it easier to solve for the missing information. Understanding the formula is crucial because it allows us to calculate the length of the line segments by leveraging the coordinates, something that would not be possible without the formula. Knowing this formula is not just beneficial for this problem but also for numerous other geometry problems. Now that we understand the distance formula and have identified its importance, let's go ahead and work on solving the actual problem.
Applying the Distance Formula
Alright, now that we have all the groundwork laid out, let's get down to the nitty-gritty and apply the distance formula to our problem. We know the coordinates of the vertices of the quadrilateral: A(-4, 1), B(-1, 3), C(3, 1), and D(0, -1). We want to find the length of CD. So, we'll use the coordinates of points C and D in the distance formula. Let's label the coordinates: Point C (3, 1) as (x1, y1), and Point D (0, -1) as (x2, y2). Now we'll substitute these values into the distance formula: Distance = √((x2 - x1)² + (y2 - y1)²). This step is where we replace the variables in the formula with the actual numbers from our coordinates. It is super important to ensure we plug in the correct coordinates. Substituting these values gives us: Distance = √((0 - 3)² + (-1 - 1)²). See? Not so bad, right? We've just plugged in the numbers, and now we are ready to move on to the next step which involves doing the math. We'll simplify the formula to find the difference of the x-coordinates and y-coordinates.
Now that we have plugged in the coordinates into the formula, let's go ahead and calculate the distance between the points C and D. Continuing with our calculations: Distance = √((-3)² + (-2)²). Simplifying further, we square the numbers inside the parenthesis. This step involves calculating the squares of the numbers obtained. -3 squared is 9, and -2 squared is 4, so it becomes Distance = √(9 + 4). Then, we add the two numbers together. We can now simplify this to: Distance = √(13). And now comes the final step. We take the square root of 13. At this stage, you'll probably want to use a calculator. The square root of 13 is approximately 3.60555. That's our answer, which gives us the length of the line segment CD. However, the problem asks us to round the answer to the nearest tenth. So, let's do that. Rounding 3.60555 to the nearest tenth gives us 3.6. So, the length of CD is approximately 3.6 units. It's a fantastic feeling to arrive at the solution after all the steps! Congratulations, guys. You've successfully calculated the length of CD using the distance formula!
Selecting the Correct Answer
Okay, awesome job, everyone! We've found the length of CD, and now we need to see how it matches up with the multiple-choice options we were given. The options were: a. 3.6 b. 4.5 c. 5.4 d. 2.8. Our calculated answer was 3.6. Luckily, that exact value is one of the answer choices. So, by matching our calculated answer with the options, we know we have the correct answer. The correct answer is a. 3.6. This is a great example of how understanding the distance formula can help solve geometry problems. By going through the steps, we were able to find the length of the line segment CD and then confirm that the result matched the given choices. This process of calculation and then comparison is a typical way of solving problems and making sure you are on the right track. Remember, the key is to understand the concepts, practice the formulas, and break down complex problems into smaller, more manageable steps. Don't be afraid to double-check your work and to use a calculator when needed. That helps ensure that you can be confident when choosing the correct answer. Knowing the right answer helps you feel more confident about your understanding of the concept. It also helps in improving your overall problem-solving skills, not only in math but in other areas too.
Conclusion
So, there you have it, folks! We've successfully found the length of CD using the distance formula. We started with the basic coordinates, learned the distance formula, plugged in the numbers, and crunched some numbers to get our answer. Remember, the distance formula is a powerful tool for solving geometry problems. Keep practicing, and you'll become a pro in no time. This is not just about solving one math problem. It’s about building a solid foundation in geometry, learning a new problem-solving skill, and enhancing your mathematical intuition. So keep up the great work, and remember that with practice and understanding, you can conquer any math challenge that comes your way. Keep practicing and applying these concepts. You've got this!