The Distance Between The Points { (x_1, Y_1)$}$ And { (4,8)$}$ Is The Square Root Of { (x_1 - 4)^2 + (y_1 - 8)^2$}$.A. True B. False

Understanding the Concept of Distance in Mathematics

In mathematics, the distance between two points in a coordinate plane is a fundamental concept that is used to measure the length of a line segment connecting two points. The distance formula is a mathematical expression that calculates the distance between two points in a coordinate plane. In this article, we will explore the concept of distance between two points in a coordinate plane and how it is calculated using the distance formula.

The Distance Formula

The distance formula is a mathematical expression that calculates the distance between two points in a coordinate plane. The formula is given by:

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

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

Applying the Distance Formula to the Given Problem

In the given problem, we are asked to find the distance between the points $(x_1, y_1)$ and $(4, 8)$. Using the distance formula, we can calculate the distance as follows:

$$d = \sqrt{(4 - x_1)^2 + (8 - y_1)^2}$$

This is the correct formula for calculating the distance between the two points.

Evaluating the Given Statement

The given statement claims that the distance between the points $(x_1, y_1)$ and $(4, 8)$ is the square root of $(x_1 - 4)^2 + (y_1 - 8)^2$. Let's evaluate this statement by comparing it with the correct formula.

$$\sqrt{(x_1 - 4)^2 + (y_1 - 8)^2} = \sqrt{(4 - x_1)^2 + (8 - y_1)^2}$$

As we can see, the two expressions are equivalent. Therefore, the given statement is TRUE.

Conclusion

In conclusion, the distance between the points $(x_1, y_1)$ and $(4, 8)$ is indeed the square root of $(x_1 - 4)^2 + (y_1 - 8)^2$. This is a fundamental concept in mathematics that is used to measure the length of a line segment connecting two points in a coordinate plane. The distance formula is a mathematical expression that calculates the distance between two points in a coordinate plane, and it is used extensively in various fields such as physics, engineering, and computer science.

Real-World Applications of the Distance Formula

The distance formula has numerous real-world applications in various fields such as:

• Physics: The distance formula is used to calculate the distance between two objects in a coordinate plane, which is essential in understanding the motion of objects.
• Engineering: The distance formula is used to calculate the distance between two points in a coordinate plane, which is essential in designing and building structures such as bridges, roads, and buildings.
• Computer Science: The distance formula is used to calculate the distance between two points in a coordinate plane, which is essential in developing algorithms for tasks such as image processing and computer vision.

Common Mistakes to Avoid

When using the distance formula, there are several common mistakes to avoid:

• Incorrectly ordering the coordinates: Make sure to order the coordinates correctly, i.e., $(x_1, y_1)$ and $(x_2, y_2)$.
• Incorrectly calculating the differences: Make sure to calculate the differences correctly, i.e., $(x_2 - x_1)$ and $(y_2 - y_1)$.
• Incorrectly squaring the differences: Make sure to square the differences correctly, i.e., $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$.

Tips and Tricks

Here are some tips and tricks to help you master the distance formula:

• Practice, practice, practice: The more you practice using the distance formula, the more comfortable you will become with it.
• Use a calculator: If you are having trouble calculating the distance manually, use a calculator to help you.
• Check your work: Always check your work to ensure that you have calculated the distance correctly.

Conclusion

Q: What is the distance formula?

A: The distance formula is a mathematical expression that calculates 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?

A: To use the distance formula, you need to know the coordinates of the two points. Let's say you want to find the distance between the points $(x_1, y_1)$ and $(x_2, y_2)$. Simply plug in the coordinates into the formula and calculate the distance.

Q: What are the coordinates of the two points?

A: The coordinates of the two points are $(x_1, y_1)$ and $(x_2, y_2)$. For example, if the two points are $(2, 3)$ and $(4, 5)$, then the coordinates are $(2, 3)$ and $(4, 5)$.

Q: How do I calculate the differences in the coordinates?

A: To calculate the differences in the coordinates, simply subtract the corresponding coordinates. For example, if the two points are $(2, 3)$ and $(4, 5)$, then the differences are $(4 - 2) = 2$ and $(5 - 3) = 2$.

Q: How do I square the differences?

A: To square the differences, simply multiply the differences by themselves. For example, if the differences are $2$ and $2$, then the squared differences are $2^2 = 4$ and $2^2 = 4$.

Q: How do I calculate the distance?

A: To calculate the distance, simply plug in the squared differences into the distance formula and calculate the square root.

Q: What is the unit of measurement for the distance?

A: The unit of measurement for the distance is typically measured in units of length, such as meters, feet, or inches.

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

A: Yes, you can use the distance formula to find the distance between two points in 3D space. However, you will 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: Can I use the distance formula to find the distance between two points on a sphere?

A: Yes, you can use the distance formula to find the distance between two points on a sphere. However, you will need to use the spherical distance formula, which is given by:

$$d = \arccos(\sin(\phi_1)\sin(\phi_2) + \cos(\phi_1)\cos(\phi_2)\cos(\theta_2 - \theta_1))$$

where $\phi$ is the latitude and $\theta$ is the longitude.

Q: What are some common mistakes to avoid when using the distance formula?

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

• Incorrectly ordering the coordinates: Make sure to order the coordinates correctly, i.e., $(x_1, y_1)$ and $(x_2, y_2)$.
• Incorrectly calculating the differences: Make sure to calculate the differences correctly, i.e., $(x_2 - x_1)$ and $(y_2 - y_1)$.
• Incorrectly squaring the differences: Make sure to square the differences correctly, i.e., $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$.

Q: How can I practice using the distance formula?

A: You can practice using the distance formula by working through examples and exercises. You can also use online resources, such as calculators and interactive tools, to help you practice.