Select The Correct Answer.The Endpoints Of W X ‾ \overline{WX} W X Are W ( 5 , − 3 W(5, -3 W ( 5 , − 3 ] And X ( − 1 , − 9 X(-1, -9 X ( − 1 , − 9 ].What Is The Length Of W X ‾ \overline{WX} W X ?A. 6 B. 12 C. 16 D. 2 3 2 \sqrt{3} 2 3 ​ E. 6 2 6 \sqrt{2} 6 2 ​

Introduction

In geometry, the length of a line segment can be calculated using the distance formula. This formula is essential in various mathematical applications, including graphing, geometry, and trigonometry. In this article, we will explore how to calculate the length of a line segment in a coordinate plane using the distance formula.

The Distance Formula

The distance formula is a mathematical formula used to find 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 $d$ is the distance between the two points, and $(x_1, y_1)$ 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 find the length of $\overline{WX}$, we can use the distance formula.

First, we need to identify the coordinates of the two points. The coordinates of point $W$ are $(5, -3)$, and the coordinates of point $X$ are $(-1, -9)$.

Next, we can plug these values into the distance formula:

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

Simplifying the expression, we get:

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

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

$$d = \sqrt{72}$$

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

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

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

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

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

Conclusion

In this article, we have learned how to calculate the length of a line segment in a coordinate plane using the distance formula. We have applied this formula to find the length of $\overline{WX}$, which is $6 \sqrt{2}$. This formula is essential in various mathematical applications, including graphing, geometry, and trigonometry.

Answer

The correct answer is E. $6 \sqrt{2}$.

Additional Examples

Here are some additional examples of calculating the length of a line segment using the distance formula:

• Find the length of $\overline{AB}$, where $A(2, 3)$ and $B(4, 5)$.
• Find the length of $\overline{CD}$, where $C(-2, 1)$ and $D(3, -2)$.
• Find the length of $\overline{EF}$, where $E(1, -1)$ and $F(-2, 4)$.

Solutions

• The length of $\overline{AB}$ is $\sqrt{(4 - 2)^2 + (5 - 3)^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2 \sqrt{2}$.
• The length of $\overline{CD}$ is $\sqrt{(3 - (-2))^2 + (-2 - 1)^2} = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}$.
• The length of $\overline{EF}$ is $\sqrt{(-2 - 1)^2 + (4 - (-1))^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}$.

Conclusion

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Q: What is the distance formula?

A: The distance formula is a mathematical formula used to find 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 find the length of a line segment?

A: To use the distance formula, you need to identify the coordinates of the two points and plug them into the formula. The coordinates of the two points are represented by $(x_1, y_1)$ and $(x_2, y_2)$. The distance formula is then used to find the distance between these two points.

Q: What are the coordinates of the two points?

A: The coordinates of the two points are the x and y values that represent the location of the points on the coordinate plane. For example, if the coordinates of point A are (2, 3), then the x value is 2 and the y value is 3.

Q: How do I simplify the expression in the distance formula?

A: To simplify the expression in the distance formula, you need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponents (such as squaring or cubing).
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: What is the final answer in the distance formula?

A: The final answer in the distance formula is the distance between the two points, represented by the variable d.

Q: Can I use the distance formula to find the length of a line segment in a 3D coordinate system?

A: Yes, you can use the distance formula to find the length of a line segment in a 3D coordinate system. 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 are some real-world applications of the distance formula?

A: The distance formula has many real-world applications, including:

• Navigation: The distance formula is used in navigation systems to find the distance between two points on a map.
• Geography: The distance formula is used in geography to find the distance between two points on the Earth's surface.
• Physics: The distance formula is used in physics to find the distance traveled by an object.
• Engineering: The distance formula is used in engineering to find the distance between two points in a design.

Q: Can I use the distance formula to find the length of a line segment in a non-coordinate system?

A: No, the distance formula is only applicable to coordinate systems. If you are working with a non-coordinate system, you need to use a different method to find the length of a line segment.

Conclusion

In conclusion, the distance formula is a powerful tool for calculating the length of a line segment in a coordinate plane. By understanding how to use the distance formula, you can find the length of any line segment, given its endpoints. We have answered some frequently asked questions about the distance formula and provided additional information to reinforce our understanding of this concept.