Find The Distance Between The Two Points: \[$(0, -2)\$\] And \[$(5, -7)\$\].If Your Solution Appears As \[$5 \sqrt{2}\$\], Write It As 5 Sqrt 2.Distance \[$=\$\] \[$\qquad\$\]Blank 1: \[$\square\$\]

by ADMIN 201 views

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 learn how to find the distance between two points in a coordinate plane using the distance formula.

The Distance Formula

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

d=(x2βˆ’x1)2+(y2βˆ’y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points.

Example: Finding the Distance Between Two Points

Let's consider an example to find the distance between two points in a coordinate plane. We are given two points: (0,βˆ’2)(0, -2) and (5,βˆ’7)(5, -7). We need to find the distance between these two points.

To find the distance, we will use the distance formula:

d=(x2βˆ’x1)2+(y2βˆ’y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points.

Substituting the values of the coordinates, we get:

d=(5βˆ’0)2+(βˆ’7βˆ’(βˆ’2))2d = \sqrt{(5 - 0)^2 + (-7 - (-2))^2}

d=(5)2+(βˆ’5)2d = \sqrt{(5)^2 + (-5)^2}

d=25+25d = \sqrt{25 + 25}

d=50d = \sqrt{50}

d=25Γ—2d = \sqrt{25 \times 2}

d=52d = 5 \sqrt{2}

Therefore, the distance between the two points (0,βˆ’2)(0, -2) and (5,βˆ’7)(5, -7) is 525 \sqrt{2}.

Discussion

The distance formula is a powerful tool that can be used to find the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is used in various fields such as geometry, trigonometry, and physics.

In this article, we learned how to find the distance between two points in a coordinate plane using the distance formula. We also saw an example of how to use the distance formula to find the distance between 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. It is derived from the Pythagorean theorem and is used in various fields such as geometry, trigonometry, and physics. We learned how to use the distance formula to find the distance between two points in a coordinate plane and saw an example of how to use it.

Applications of the Distance Formula

The distance formula has many applications in real-life situations. Some of the applications of the distance formula include:

  • Navigation: The distance formula can be used to find the distance between two locations on a map.
  • Surveying: The distance formula can be used to find the distance between two points on a survey map.
  • Physics: The distance formula can be used to find the distance between two objects in motion.
  • Engineering: The distance formula can be used to find the distance between two points in a design or blueprint.

Tips and Tricks

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

  • Make sure to substitute the values of the coordinates correctly.
  • Use the correct order of operations.
  • Simplify the expression as much as possible.
  • Check your answer to make sure it is reasonable.

Practice Problems

Here are some practice problems to help you practice using the distance formula:

  • Find the distance between the points (3,4)(3, 4) and (6,8)(6, 8).
  • Find the distance between the points (βˆ’2,5)(-2, 5) and (3,βˆ’1)(3, -1).
  • Find the distance between the points (0,0)(0, 0) and (4,3)(4, 3).

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. It is derived from the Pythagorean theorem and is used in various fields such as geometry, trigonometry, and physics. We learned how to use the distance formula to find the distance between two points in a coordinate plane and saw an example of how to use it. We also discussed some of the applications of the distance formula and provided some tips and tricks to help you use it.

Introduction

In our previous article, we learned how to find the distance between two points in a coordinate plane using the distance formula. In this article, we will answer some frequently asked questions about the distance formula.

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. It is given by:

d=(x2βˆ’x1)2+(y2βˆ’y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points.

Q: How do I use the distance formula?

A: To use the distance formula, you need to substitute the values of the coordinates into the formula and simplify the expression. Here's an example:

Let's say we want to find the distance between the points (3,4)(3, 4) and (6,8)(6, 8). We would substitute the values of the coordinates into the formula as follows:

d=(6βˆ’3)2+(8βˆ’4)2d = \sqrt{(6 - 3)^2 + (8 - 4)^2}

d=(3)2+(4)2d = \sqrt{(3)^2 + (4)^2}

d=9+16d = \sqrt{9 + 16}

d=25d = \sqrt{25}

d=5d = 5

Therefore, the distance between the two points is 55.

Q: What if the coordinates are negative?

A: If the coordinates are negative, you can simply substitute the values into the formula and simplify the expression. Here's an example:

Let's say we want to find the distance between the points (βˆ’2,5)(-2, 5) and (3,βˆ’1)(3, -1). We would substitute the values of the coordinates into the formula as follows:

d=(3βˆ’(βˆ’2))2+(βˆ’1βˆ’5)2d = \sqrt{(3 - (-2))^2 + (-1 - 5)^2}

d=(5)2+(βˆ’6)2d = \sqrt{(5)^2 + (-6)^2}

d=25+36d = \sqrt{25 + 36}

d=61d = \sqrt{61}

Therefore, the distance between the two points is 61\sqrt{61}.

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 need to use the 3D distance formula, which is given by:

d=(x2βˆ’x1)2+(y2βˆ’y1)2+(z2βˆ’z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

where dd is the distance between the two points, (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2) are the coordinates of the two points.

Q: What if I have a point and a line, and I want to find the distance between the point and the line?

A: If you have a point and a line, and you want to find the distance between the point and the line, you can use the formula for the distance between a point and a line. The formula is given by:

d=∣Ax+By+C∣A2+B2d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}

where dd is the distance between the point and the line, (x,y)(x, y) is the point, and Ax+By+C=0Ax + By + C = 0 is the equation of the line.

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

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

d=(x2βˆ’x1)2+(y2βˆ’y1)2βˆ’r2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 - r^2}

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points, and rr is the radius of the circle.

Conclusion

In conclusion, the distance formula is a powerful tool that can be used to find the distance between two points in a coordinate plane. We answered some frequently asked questions about the distance formula and provided some examples of how to use it. We also discussed some of the applications of the distance formula and provided some tips and tricks to help you use it.