The Endpoints Of $\overline{WX}$ Are $W(5, -3)$ And \$X(-1, -9)$[/tex\].What Is The Length Of $\overline{WX}$?

Introduction

In geometry, a line segment is a part of a line that is bounded by two distinct points. The length of a line segment can be calculated using the distance formula, which is derived from the Pythagorean theorem. In this article, we will explore how to calculate the length of a line segment given its endpoints.

The Distance Formula

The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. It is given by:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Calculating the Length of $\overline{WX}$

We are given the endpoints of $\overline{WX}$ as $W(5, -3)$ and $X(-1, -9)$. To calculate the length of $\overline{WX}$, we can use the distance formula.

Step 1: Identify the Coordinates of the Endpoints

The coordinates of the endpoints are given as:

• $$W(5, -3)$$

• $$X(-1, -9)$$

Step 2: Plug in the Coordinates into the Distance Formula

We can now plug in the coordinates into the distance formula:

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

Step 3: Simplify the Expression

Simplifying the expression, we get:

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

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

$$d = \sqrt{72}$$

Step 4: Simplify the Square Root

The square root of 72 can be simplified as:

$$\sqrt{72} = \sqrt{36 \times 2}$$

$$\sqrt{72} = 6\sqrt{2}$$

Conclusion

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

Real-World Applications

The distance formula has numerous real-world applications, including:

• Calculating the distance between two cities
• Determining the length of a road or highway
• Measuring the distance between two points on a map
• Calculating the distance between two objects in a 3D space

Tips and Tricks

When using the distance formula, it's essential to:

• Make sure to plug in the correct coordinates
• Simplify the expression to avoid errors
• Use the correct units (e.g., meters, kilometers, etc.)

Common Mistakes

Some common mistakes to avoid when using the distance formula include:

• Forgetting to square the differences between the coordinates
• Not simplifying the expression
• Using the wrong units

Final Thoughts

In conclusion, the distance formula is a powerful tool for calculating the length of a line segment given its endpoints. By following the steps outlined in this article, you can calculate the length of a line segment with ease. Remember to always simplify the expression and use the correct units to ensure accurate results.

Introduction

In our previous article, we explored how to calculate the length of a line segment given its endpoints using the distance formula. In this article, we will answer some frequently asked questions related to the distance formula and calculating the length of a line segment.

Q&A

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. It is given by:

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

Q: How do I use the distance formula to calculate the length of a line segment?

A: To use the distance formula, you need to:

1. Identify the coordinates of the endpoints of the line segment.
2. Plug in the coordinates into the distance formula.
3. Simplify the expression to get the final answer.

Q: What if the coordinates are negative?

A: If the coordinates are negative, you can simply plug them into the distance formula as is. The formula will handle the negative values correctly.

Q: Can I use the distance formula to calculate the distance between two points in 3D space?

A: Yes, you can use the distance formula to calculate the distance between two points in 3D space. However, you need to use the 3D distance formula, which is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

Q: What if I have a line segment with a negative length?

A: If you have a line segment with a negative length, it means that the line segment is not a valid line segment. In other words, the two points are not connected by a line segment.

Q: Can I use the distance formula to calculate the length of a line segment with a slope?

A: Yes, you can use the distance formula to calculate the length of a line segment with a slope. However, you need to use the slope-intercept form of the line, which is given by:

$$y = mx + b$$

where m is the slope and b is the y-intercept.

Q: What if I have a line segment with a slope and a negative length?

A: If you have a line segment with a slope and a negative length, it means that the line segment is not a valid line segment. In other words, the two points are not connected by a line segment.

Tips and Tricks

When using the distance formula, it's essential to:

• Make sure to plug in the correct coordinates
• Simplify the expression to avoid errors
• Use the correct units (e.g., meters, kilometers, etc.)

Common Mistakes

Some common mistakes to avoid when using the distance formula include:

• Forgetting to square the differences between the coordinates
• Not simplifying the expression
• Using the wrong units

Final Thoughts

In conclusion, the distance formula is a powerful tool for calculating the length of a line segment given its endpoints. By following the steps outlined in this article and avoiding common mistakes, you can calculate the length of a line segment with ease.

Additional Resources

For more information on the distance formula and calculating the length of a line segment, check out the following resources:

• Distance Formula Wikipedia Article
• Calculating the Length of a Line Segment
• Distance Formula Calculator

Conclusion

In conclusion, the distance formula is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to use the distance formula, you can calculate the length of a line segment with ease and apply it to various fields such as physics, engineering, and computer science.