Calculating The Length Of Line Segment WX A Step-by-Step Guide

In the realm of coordinate geometry, determining the length of a line segment is a fundamental concept. This article delves into the process of calculating the length of a line segment, specifically focusing on the line segment $\overline{WX}$ with endpoints $W(5,-3)$ and $X(-1,-9)$. We will explore the distance formula, a powerful tool used to compute the distance between two points in a coordinate plane. By applying this formula, we will systematically calculate the length of $\overline{WX}$ and arrive at the correct answer. Understanding the distance formula and its applications is crucial for various mathematical and real-world problems, making this a valuable topic to master.

Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem and provides a method for calculating the distance between two points in a coordinate plane. Given two points, $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between them is given by:

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

This formula essentially calculates the length of the hypotenuse of a right triangle formed by the horizontal and vertical distances between the two points. The horizontal distance is represented by $|x_2 - x_1|$, and the vertical distance is represented by $|y_2 - y_1|$. Squaring these distances, summing them, and then taking the square root gives us the distance between the two points. The distance formula is a cornerstone of coordinate geometry, finding applications in various fields, including navigation, engineering, and computer graphics.

Applying the Distance Formula to Find the Length of WX

To find the length of $\overline{WX}$, we will apply the distance formula to the coordinates of points $W(5,-3)$ and $X(-1,-9)$. Let's identify the coordinates:

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

Now, we substitute these values into the distance formula:

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

Simplifying the expression inside the square root:

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

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

$$d = \sqrt{72}$$

Further simplifying the square root:

$$d = \sqrt{36 \times 2}$$

$$d = 6\sqrt{2}$$

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

Analyzing the Answer Choices

Now that we have calculated the length of $\overline{WX}$ to be $6\sqrt{2}$, let's examine the given answer choices:

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

Comparing our calculated length with the answer choices, we can clearly see that option E, $6 \sqrt{2}$, matches our result. Therefore, the correct answer is E.

Visualizing the Line Segment and Distance

To further solidify our understanding, let's visualize the line segment $\overline{WX}$ on a coordinate plane. Plotting the points $W(5,-3)$ and $X(-1,-9)$, we can draw a line segment connecting them. The distance we calculated, $6\sqrt{2}$, represents the straight-line distance between these two points. This visualization helps to connect the abstract concept of the distance formula with a tangible geometric representation.

Alternative Methods for Finding the Length

While the distance formula is the most direct method for finding the length of a line segment given its endpoints, there are alternative approaches that can be used. One such method involves using the Pythagorean theorem directly. As mentioned earlier, the distance formula is derived from the Pythagorean theorem. By visualizing a right triangle with $\overline{WX}$ as the hypotenuse, we can find the lengths of the horizontal and vertical legs and then apply the Pythagorean theorem to find the length of the hypotenuse. This alternative method provides a deeper understanding of the relationship between the distance formula and the Pythagorean theorem.

Real-World Applications of Distance Calculation

The concept of calculating distances between points has numerous applications in real-world scenarios. In navigation, GPS systems use the distance formula to determine the distance between locations. In engineering, it is used in structural design and surveying. In computer graphics, it is used in calculating distances between objects in a virtual environment. Understanding distance calculation is therefore essential in various fields and professions.

Common Mistakes to Avoid

When applying the distance formula, it is important to avoid common mistakes that can lead to incorrect results. One common mistake is incorrect substitution of coordinates into the formula. Ensure that the $x$ and $y$ coordinates are correctly placed in the formula. Another common mistake is errors in arithmetic, particularly when dealing with negative numbers and square roots. It is crucial to double-check calculations to ensure accuracy. By being aware of these potential pitfalls, you can minimize the chances of making mistakes.

Practice Problems

To reinforce your understanding of the distance formula and its application, it is essential to practice solving problems. Here are a few practice problems:

1. Find the distance between the points $A(2, 3)$ and $B(5, 7)$.
2. Find the distance between the points $C(-1, 4)$ and $D(3, -2)$.
3. Find the distance between the points $E(0, -5)$ and $F(4, 0)$.

By working through these problems, you will gain confidence in your ability to apply the distance formula and calculate the length of line segments.

Conclusion

In this comprehensive guide, we have explored the process of calculating the length of a line segment using the distance formula. We have seen how the distance formula is derived from the Pythagorean theorem and how it can be applied to find the distance between two points in a coordinate plane. By applying the distance formula to the endpoints of $\overline{WX}$, $W(5,-3)$ and $X(-1,-9)$, we found that the length of $\overline{WX}$ is $6\sqrt{2}$. We also discussed alternative methods for finding the length, real-world applications of distance calculation, common mistakes to avoid, and provided practice problems to further enhance your understanding. Mastering the distance formula is a valuable skill that will serve you well in various mathematical and real-world contexts.