Calculate The Length Of Line Segment WX Endpoints W(5,-3) And X(-1,-9)

In mathematics, determining the length of a line segment is a fundamental concept. When given the coordinates of the endpoints of a line segment, the distance formula becomes an invaluable tool. This article delves into the process of calculating the length of line segment ${\overline{WX}}$, where the coordinates of point W are (5, -3) and the coordinates of point X are (-1, -9). We will explore the distance formula, its application, and arrive at the solution while explaining each step in detail. This comprehensive guide will provide you with a solid understanding of how to apply the distance formula to solve similar problems. This concept is crucial in various mathematical contexts, including geometry, coordinate geometry, and even in practical applications such as navigation and engineering. Mastering the distance formula is an essential step in building a strong foundation in mathematics.

Understanding the Distance Formula

To accurately calculate the length of line segment ${\overline{WX}}$, we must first understand and apply the distance formula. The distance formula is derived from the Pythagorean theorem and is used to find the distance between two points in a coordinate plane. If we have two points, say ${(x_1, y_1)}$ and ${(x_2, y_2)}$, the distance ${d}$ between these points is given by:

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

This formula calculates the straight-line distance between the two points by considering the difference in their x-coordinates and the difference in their y-coordinates. The squaring of the differences ensures that we are dealing with positive values, and the square root gives us the actual distance. The distance formula is a cornerstone of coordinate geometry, allowing us to quantitatively analyze geometric figures and their properties. It transforms geometric problems into algebraic equations, making them easier to solve. Understanding the derivation of this formula from the Pythagorean theorem provides a deeper insight into its application. It's essential to remember that the order in which you subtract the coordinates does not matter because the differences are squared, eliminating any negative signs. This formula is not only used in academic settings but also has practical applications in fields like computer graphics, geographic information systems (GIS), and various engineering disciplines.

Applying the Distance Formula to ${\overline{WX}}$

In our specific case, the endpoints of line segment ${\overline{WX}}$ are given as W(5, -3) and X(-1, -9). To apply the distance formula, we can identify the coordinates as follows:

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

Now, we substitute these values into the distance formula:

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

This substitution is a critical step in solving the problem. It involves carefully placing the coordinates into their respective positions in the formula. A common mistake is to mix up the x and y values or to incorrectly handle the negative signs. To avoid these errors, it's helpful to write the formula down first and then systematically substitute the values. This ensures that each coordinate is correctly placed, minimizing the chances of making a mistake. The distance formula essentially calculates the hypotenuse of a right-angled triangle formed by the difference in x-coordinates and the difference in y-coordinates. This geometric interpretation can be helpful in visualizing the formula and understanding its application. By carefully substituting the values, we set the stage for the next step, which involves simplifying the expression and calculating the final distance.

Calculating the Length

After substituting the coordinates into the distance formula, we simplify the expression step by step:

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

First, perform the subtractions inside the parentheses:

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

Next, square the numbers:

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

Then, add the results:

${ d = \sqrt{72} }$

Now, simplify the square root. We look for perfect square factors of 72. We can rewrite 72 as 36 * 2:

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

Finally, take the square root of 36:

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

Therefore, the length of line segment ${\overline{WX}}$ is ${6\sqrt{2}}$. Each step in this calculation is crucial. The order of operations must be followed correctly to arrive at the correct answer. Squaring the negative numbers results in positive values, which is a common point of error if not handled carefully. Simplifying the square root involves finding the largest perfect square factor, which can sometimes be challenging. Breaking down the number into its prime factors can be a helpful strategy for simplifying square roots. The final result, ${6\sqrt{2}}$, represents the exact distance between the two points. This value can also be approximated to a decimal for practical purposes, but the exact form is often preferred in mathematical contexts.

Solution and Answer

The calculated length of line segment ${\overline{WX}}$ is ${6\sqrt{2}}$ units. Therefore, the correct answer is:

• E. ${6 \sqrt{2}}$

This solution demonstrates the application of the distance formula and the simplification of square roots. The distance formula is a fundamental tool in coordinate geometry, and understanding how to use it is essential for solving a variety of problems. This specific example highlights the importance of careful calculation and simplification to arrive at the correct answer. The ability to apply this formula accurately is crucial for students and professionals alike in fields that require spatial reasoning and geometric analysis. By understanding the underlying principles and practicing the application of the formula, one can confidently solve similar problems in various contexts. The correct answer, ${6\sqrt{2}}$, represents the precise distance between the two points, providing a clear and concise solution to the problem.

In summary, this article has provided a step-by-step guide to calculating the length of line segment ${\overline{WX}}$ using the distance formula. We started by understanding the formula, applied it to the given coordinates, and simplified the result to arrive at the final answer. This comprehensive approach ensures a solid understanding of the process and its application in similar problems. Understanding the distance formula is crucial for various mathematical contexts, including geometry, coordinate geometry, and practical applications such as navigation and engineering. Mastering the distance formula is an essential step in building a strong foundation in mathematics. This skill is not only valuable for academic purposes but also has real-world applications in various fields. By following the steps outlined in this article, you can confidently calculate the distance between any two points in a coordinate plane.