What Is The Distance Between The Points ( − 4 , 2 (-4,2 ( − 4 , 2 ] And ( 3 , − 5 (3,-5 ( 3 , − 5 ]? D = ( X 2 − X 1 ) 2 + ( Y 2 − Y 1 ) 2 D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} D = ( X 2 − X 1 ) 2 + ( Y 2 − Y 1 ) 2 A. 10 \sqrt{10} 10 B. 28 \sqrt{28} 28 C. 98 \sqrt{98} 98 D. 117 \sqrt{117} 117

Mar 13, 2025 by ADMIN 329 views

**What is the Distance Between Two Points in a Coordinate Plane?**

Introduction

In mathematics, the distance between two points in a coordinate plane is a fundamental concept that is used in various fields such as geometry, trigonometry, and physics. The distance between two points can be calculated using the distance formula, which is derived from the Pythagorean theorem. In this article, we will discuss the distance formula and how to use it to find the distance between two points in a coordinate plane.

The Distance Formula

The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane. The formula is as follows:

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

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.

How to Use the Distance Formula

To use the distance formula, you need to follow these steps:

Identify the coordinates of the two points. In this case, the coordinates of the two points are $(-4, 2)$ and $(3, -5)$ .
Plug the coordinates into the distance formula. In this case, the distance formula becomes:

$d = \sqrt{\left(3 - (-4)\right)^2 + \left(-5 - 2\right)^2}$ 3. Simplify the equation. In this case, the equation becomes:

$d = \sqrt{\left(7\right)^2 + \left(-7\right)^2}$ 4. Calculate the square of the numbers. In this case, the equation becomes:

$d = \sqrt{49 + 49}$ 5. Add the numbers. In this case, the equation becomes:

$d = \sqrt{98}$ 6. Take the square root of the number. In this case, the distance between the two points is:

$d = \sqrt{98}$

Conclusion

In conclusion, the distance between two points in a coordinate plane can be calculated using the distance formula. The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane. To use the distance formula, you need to identify the coordinates of the two points, plug the coordinates into the distance formula, simplify the equation, calculate the square of the numbers, add the numbers, and take the square root of the number.

Answer

The correct answer is:

C. $\sqrt{98}$

Discussion

The distance formula is a fundamental concept in mathematics that is used in various fields such as geometry, trigonometry, and physics. The distance formula is used to find the distance between two points in a coordinate plane. The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane.

Example Problems

Here are some example problems that you can use to practice using the distance formula:

Find the distance between the points $(2, 3)$ and $(4, 5)$ .
Find the distance between the points $(-2, 1)$ and $(3, -4)$ .
Find the distance between the points $(1, 2)$ and $(-3, 4)$ .

Tips and Tricks

Here are some tips and tricks that you can use to help you remember the distance formula:

The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane.
The distance formula is derived from the Pythagorean theorem.
The distance formula is used to find the distance between two points in a coordinate plane.
The distance formula is a fundamental concept in mathematics that is used in various fields such as geometry, trigonometry, and physics.

Conclusion

Q: What is the distance formula?

A: The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane. The formula is as follows:

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

Q: How do I use the distance formula?

A: To use the distance formula, you need to follow these steps:

Identify the coordinates of the two points.
Plug the coordinates into the distance formula.
Simplify the equation.
Calculate the square of the numbers.
Add the numbers.
Take the square root of the number.

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 $(-4, 2)$ and $(3, -5)$ , then the coordinates are $(-4, 2)$ and $(3, -5)$ .

Q: How do I plug the coordinates into the distance formula?

A: To plug the coordinates into the distance formula, you need to substitute the values of $x_1$ , $y_1$ , $x_2$ , and $y_2$ into the formula. For example, if the coordinates are $(-4, 2)$ and $(3, -5)$ , then the distance formula becomes:

$d = \sqrt{\left(3 - (-4)\right)^2 + \left(-5 - 2\right)^2}$

Q: How do I simplify the equation?

A: To simplify the equation, you need to follow the order of operations (PEMDAS):

Evaluate the expressions inside the parentheses.
Exponentiate the numbers.
Multiply and divide the numbers from left to right.
Add and subtract the numbers from left to right.

Q: What is the final answer?

A: The final answer is the value of $d$ , which is the distance between the two points.

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. The formula is as follows:

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

Q: What are some real-world applications of the distance formula?

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

Calculating the distance between two cities or towns.
Finding the distance between two points on a map.
Calculating the distance between two objects in space.
Determining the distance between two points on a coordinate grid.

Q: Can I use the distance formula to find the distance between two points on a coordinate grid?

A: Yes, you can use the distance formula to find the distance between two points on a coordinate grid. The formula is the same as the one used to find the distance between two points in 2D space.

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:

Not following the order of operations (PEMDAS).
Not simplifying the equation correctly.
Not taking the square root of the number correctly.
Not using the correct coordinates of the two points.

Conclusion

In conclusion, the distance formula is a fundamental concept in mathematics that is used to find the distance between two points in a coordinate plane. The distance formula is a mathematical formula that is used to find the distance between two points in a coordinate plane. To use the distance formula, you need to identify the coordinates of the two points, plug the coordinates into the distance formula, simplify the equation, calculate the square of the numbers, add the numbers, and take the square root of the number.