Calculate Line Segment Length: A Step-by-Step Guide

Hey everyone! Let's dive into a classic geometry problem: figuring out the length of a line segment. In this case, we're dealing with the line segment $\overline{WX}$, and we're given the coordinates of its endpoints, $W(5, -3)$ and $X(-1, -9)$. Our goal is to determine the length of this segment, and we'll go through the process step-by-step. Understanding how to calculate the distance between two points is super fundamental in geometry and has applications in various fields, from computer graphics to physics. So, let's get started and break it down in a way that's easy to follow. We'll use the distance formula, which is derived from the Pythagorean theorem, to make this calculation. It's a straightforward process, and by the end of this, you'll be able to confidently calculate the length of any line segment given its endpoints. Ready? Let's go!

The Distance Formula: Your Secret Weapon

Alright guys, before we jump into the specifics, let's get familiar with the distance formula. This formula is the key to unlocking the length of our line segment. The distance formula is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the context of coordinate geometry, we can think of the line segment as the hypotenuse of a right triangle. The other two sides are the horizontal and vertical distances between the two points. The formula itself looks like this:

$\~\$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Where:

• $(x_1, y_1)$ are the coordinates of the first point.
• $(x_2, y_2)$ are the coordinates of the second point.
• $d$ is the distance (or length) between the two points.

Basically, the formula calculates the difference in the x-coordinates, squares it, calculates the difference in the y-coordinates, squares it, adds those two squared differences, and then takes the square root of the result. This gives us the straight-line distance between the two points. It's like finding the length of the hypotenuse of a right triangle formed by the points and the axes. This is a super important formula, so make sure you understand it, and it becomes second nature.

Applying the Formula to Our Problem

Now, let's apply this formula to our specific problem. We've got the coordinates of our two points: $W(5, -3)$ and $X(-1, -9)$. We'll designate $W$ as $(x_1, y_1)$ and $X$ as $(x_2, y_2)$.

• $x_1 = 5$
• $y_1 = -3$
• $x_2 = -1$
• $y_2 = -9$

Now, substitute these values into the distance formula:

$\~\$d = \sqrt{(-1 - 5)^2 + (-9 - (-3))^2}$

Simplify the terms inside the parentheses:

$\~\$d = \sqrt{(-6)^2 + (-6)^2}$

Square the numbers:

$\~\$d = \sqrt{36 + 36}$

Add the numbers:

$\~\$d = \sqrt{72}$

Finally, simplify the square root. We can break down 72 into its prime factors: $72 = 36 \times 2$. The square root of 36 is 6, so we can simplify $\sqrt{72}$ as $6\sqrt{2}$.

Therefore, the length of the line segment $\overline{WX}$ is $6\sqrt{2}$.

Understanding the Solution

So, what does this result mean? $6\sqrt{2}$ is the exact distance between the points $W$ and $X$. If you were to approximate the value (using a calculator), you'd get roughly 8.49. This length represents the straight-line distance between the two points in the coordinate plane. It's the shortest possible path from $W$ to $X$. The fact that we got an irrational number (a number that cannot be expressed as a simple fraction) tells us that the distance isn't a whole number, nor is it a simple fraction. It's a more precise measure of the length, and in many cases, keeping the answer in the exact form ($6\sqrt{2}$) is preferable to an approximation because it preserves the accuracy of the result. This exercise reinforces how the distance formula provides a precise way to calculate distances in a coordinate system, which is fundamental to geometry and many related fields.

Analyzing the Answer Choices

Alright, now that we've calculated the length of $\overline{WX}$ to be $6\sqrt{2}$, let's take a look at the multiple-choice options provided and make sure our answer aligns perfectly. This step is crucial to confirm our calculations and ensure we select the correct response. It's also a good practice in test-taking strategy, helping us to validate our work. So, here's what we got:

A. 6 B. 12 C. 16 D. $2\sqrt{3}$ E. $6\sqrt{2}$

As we computed earlier, the correct answer is $6\sqrt{2}$. Let's break down why the other options are incorrect. Option A, 6, is a whole number, but it's not the actual distance between the points. It's possible that a calculation error could lead to this result, perhaps forgetting to square the differences or not taking the square root at the end. Option B, 12, seems to be a product of an error too. Option C, 16, is also incorrect; it's the result of adding the squares before taking the square root or some similar mistake. Option D, $2\sqrt{3}$, is not correct either, as it's a different square root that doesn't come from our distance calculations. Therefore, by a process of elimination, with the correct answer being E, $6\sqrt{2}$, which matches our calculation perfectly. This methodical checking ensures we are confident and correct in our selection.

Why Understanding the Process Matters

This whole process isn't just about getting the right answer; it's about understanding the underlying principles. Recognizing how to apply the distance formula is important. It gives you a tool to find distances in any coordinate system. Moreover, knowing how to break down a problem, like we did here, makes tackling other geometry questions easier. It encourages logical thinking, which helps in various areas of math and beyond. This approach is applicable not only to geometry problems but also to many other areas of mathematics and science where you need to find distances, relationships, or positions in space. The skills acquired, such as substituting values, simplifying equations, and understanding square roots, are fundamental to building a strong foundation in mathematics. The ability to explain your reasoning also becomes a valuable asset when you review the question or help others. It means you're not just solving problems; you're growing your mathematical knowledge and building confidence.

Common Mistakes and How to Avoid Them

Alright, let's talk about some common slip-ups that students make when dealing with the distance formula, and more importantly, how to dodge these pitfalls. Knowing what to watch out for can save you a lot of headaches and incorrect answers. Here's what to keep in mind:

• Order of Operations: The most frequent mistake involves the order of operations. Remember, you must perform the subtraction within the parentheses first, then square the result. Many students mistakenly square the individual coordinates before subtracting. Always follow the PEMDAS/BODMAS rules (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
• Sign Errors: Be very careful with the signs (positive and negative) when subtracting. A simple mistake here can lead to a totally wrong answer. When you are subtracting a negative number, remember that it's like adding a positive number. It is really easy to mess up signs, so take your time!
• Forgetting the Square Root: A lot of people forget to take the square root at the end of the calculation. Squaring the differences gives you the squares of the horizontal and vertical distances, but to get the actual distance (the length of the line segment), you must take the square root of their sum. Double-check that you've done this step.
• Simplifying Square Roots: After you take the square root, make sure you simplify it correctly. For example, if you get $\sqrt{72}$, simplify it to $6\sqrt{2}$ (as we did in our problem). If you leave it as $\sqrt{72}$, you're technically not fully answering the question.
• Incorrect Substitution: Sometimes, the coordinates get mixed up. Always make sure you correctly assign $x_1$, $x_2$, $y_1$, and $y_2$. Write them down next to the formula if that helps!

By being aware of these traps and paying close attention to each step, you'll significantly reduce your chances of making a mistake. Practice makes perfect, so work through many examples and always double-check your work. This strategy ensures that you solve the problems correctly and build your confidence in tackling these kinds of questions.

Mastering Distance Formula: Key Takeaways

Let's recap what we've learned and make sure everything sticks. The distance formula is your go-to tool for finding the length of a line segment in a coordinate plane. It's derived from the Pythagorean theorem, and it helps us find the straight-line distance between two points. By correctly substituting the coordinates, carefully performing the arithmetic, and remembering the final square root, you can calculate the distance accurately. Remember to double-check your calculations, especially those involving signs and order of operations. The practice and the ability to analyze different problems will help you get a deeper understanding. You should also understand the other options, recognizing why they're wrong helps cement your understanding. Always simplify your final answer, simplifying square roots if possible. And, most importantly, take your time, work through each step methodically, and don't be afraid to double-check your work. You're now well-equipped to solve problems involving the distance formula and can tackle similar challenges. Keep practicing, and you'll become a geometry pro in no time! This will help you in your geometry journey and in other fields too, such as computer science and physics. Great job, guys!