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# Distance Formula Worksheet, Definition, and Examples

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### Key Points about using the Distance Formula

• The distance formula is a fundamental concept in geometry that helps to calculate the distance between two points in a two-dimensional space.
• To calculate the distance between two points, one needs to know the coordinates of both points and use the distance formula.
• The distance formula is a powerful tool that can be used to solve a wide range of problems in various fields.

## How to Calculate Distance between Two Points

The distance formula is a fundamental concept in geometry that helps to calculate the distance between two points in a two-dimensional space. It is derived from the Pythagorean theorem and is widely used in various fields, including physics, engineering, and computer science. The formula is based on the principle of finding the hypotenuse of a right-angled triangle, where the two sides represent the horizontal and vertical distances between the two points.

To calculate the distance between two points, one needs to know the coordinates of both points. The distance formula involves taking the square root of the sum of the squares of the differences between the x-coordinates and the y-coordinates of the two points. The formula is expressed as follows: distance = sqrt((x2 – x1)^2 + (y2 – y1)^2). Once you have the coordinates, you can easily plug them into the formula and get the distance between the two points.

The distance formula is a powerful tool that can be used to solve a wide range of problems. It can be used to calculate the distance between two cities on a map, the length of a diagonal in a rectangle, or the distance between two points in a coordinate plane. By mastering this formula, you can gain a deeper understanding of geometry and improve your problem-solving skills.

Common Core Standard:

## How to do Distance Formula

The distance formula is a mathematical formula used to calculate the distance between two points in a space. It is used in various fields such as physics, engineering, and geometry. The formula is precise and accurate, making it an essential tool for measuring distances between objects.

To use the distance formula, you need to know the coordinates of the two points you want to measure the distance between. The formula is as follows:

Distance = √((x2 – x1)² + (y2 – y1)²)

Where x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point.

To calculate the distance between two points, you need to follow these steps:

1. Identify the coordinates of the two points.
2. Plug the coordinates into the distance formula.
3. Simplify the equation by squaring the differences between the x and y coordinates.
4. Add the squared values together.
5. Find the square root of the sum to get the distance between the two points.

It is essential to be accurate when identifying the coordinates of the two points. Any error in the coordinates will affect the accuracy of the distance calculation.

The distance formula can be used to measure the distance between any two points in a space, such as the distance between two cities, the distance between two stars, or the distance between two molecules.

In conclusion, the distance formula is a precise and accurate tool for measuring distances between two points in a space. By following the steps outlined above, you can use the formula to calculate the distance between any two points.

## Distance Formula between Two Points

Calculating the distance between two points in a coordinate plane is an essential concept in mathematics. The distance formula is a method used to calculate the distance between two points in a coordinate plane. This section will cover the different aspects of the distance formula, including the equation, how to calculate it, and the concept of line segments.

### Coordinates and Points

Before diving into the distance formula, it’s essential to understand the concept of coordinates and points. A coordinate plane is a two-dimensional plane that contains an x-axis and a y-axis. The point where the two axes intersect is called the origin, and it has coordinates (0,0). A point in a coordinate plane is represented by an ordered pair of numbers, (x,y), where x is the horizontal coordinate, and y is the vertical coordinate.

### Calculating Distance Between Two Points

To calculate the distance between two points in a coordinate plane, we need to use the distance formula. The distance formula is:

distance = √((x2 – x1)^2 + (y2 – y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

### Understanding Length and Hypotenuse

The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the case of the distance formula, the length of the hypotenuse is the distance between the two points.

### Change in X and Y

The distance formula involves finding the difference between the x-coordinates and the y-coordinates of the two points. The difference between the x-coordinates is represented by (x2 – x1), and the difference between the y-coordinates is represented by (y2 – y1). These differences are then squared and added together to find the square of the distance between the two points.

### The Concept of Line Segment

The distance formula can also be used to find the length of a line segment. A line segment is a part of a line that has two endpoints. To find the length of a line segment, we need to find the distance between the two endpoints, which can be done using the distance formula.

In summary, the distance formula is a crucial concept in mathematics that allows us to calculate the distance between two points in a coordinate plane. By understanding the coordinates and points, the length and hypotenuse, change in x and y, and the concept of line segments, we can better understand and apply the distance formula in various mathematical problems.

## Distance Formula Definition

The distance formula is a mathematical equation used to calculate the distance between two points in a coordinate plane. It is a derivation of the Pythagorean theorem and is used in analytic geometry.

The formula is expressed as:

d = √((x2 – x1)² + (y2 – y1)²)

where d represents the distance between the two points, and (x1, y1) and (x2, y2) represent the coordinates of the two points on the plane.

The distance formula can be used to determine the distance between any two points in a plane, regardless of their position or direction. It is a fundamental tool in geometry, trigonometry, and calculus.

One important thing to note is that the distance formula only works in Euclidean space, which is a type of space where the Pythagorean theorem holds true. In other types of space, such as non-Euclidean space, different formulas may be required to calculate the distance between two points.

In summary, the distance formula is a mathematical tool used to calculate the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is a fundamental tool in geometry and other branches of mathematics.

## Distance Formula Examples

The distance formula is a useful tool for finding the distance between two points in a coordinate plane. Here are a few examples of how to use the distance formula:

### Example 1: Finding the distance between two points

Suppose you want to find the distance between the points (2, 3) and (5, 7). To use the distance formula, you need to plug the coordinates of these two points into the formula:

d = sqrt((5-2)^2 + (7-3)^2)

Simplifying this equation, you get:

d = sqrt(3^2 + 4^2)

d = sqrt(9 + 16)

d = sqrt(25)

d = 5

Therefore, the distance between the points (2, 3) and (5, 7) is 5 units.

### Example 2: Finding the distance from a point to the origin

Suppose you want to find the distance from the point (4, 3) to the origin (0, 0). To use the distance formula, you need to plug the coordinates of these two points into the formula:

d = sqrt((4-0)^2 + (3-0)^2)

Simplifying this equation, you get:

d = sqrt(4^2 + 3^2)

d = sqrt(16 + 9)

d = sqrt(25)

d = 5

Therefore, the distance from the point (4, 3) to the origin (0, 0) is 5 units.

### Example 3: Finding the distance between two points in three dimensions

The distance formula can also be used to find the distance between two points in three dimensions. For example, suppose you want to find the distance between the points (1, 2, 3) and (4, 5, 6). To use the distance formula, you need to plug the coordinates of these two points into the formula:

d = sqrt((4-1)^2 + (5-2)^2 + (6-3)^2)

Simplifying this equation, you get:

d = sqrt(3^2 + 3^2 + 3^2)

d = sqrt(9 + 9 + 9)

d = sqrt(27)

d = 3sqrt(3)

Therefore, the distance between the points (1, 2, 3) and (4, 5, 6) is 3sqrt(3) units.

## Horizontal Distance Formula

The horizontal distance formula is used to calculate the distance between two points on a horizontal plane. It is one of the most basic applications of the distance formula. The formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

### Properties

The horizontal distance formula is used to calculate the distance between two points on a horizontal plane. It is useful in many fields, including architecture, engineering, and physics. The formula is based on the Pythagorean theorem, which is a fundamental concept in mathematics.

### Distance Formula

The horizontal distance formula is given by:

d = √((x₂ – x₁)² + (y₂ – y₁)²)

Where:

• d is the distance between the two points.
• x₁ and x₂ are the x-coordinates of the two points.
• y₁ and y₂ are the y-coordinates of the two points.

To use the formula, simply plug in the values for x₁, x₂, y₁, and y₂ into the formula and solve for d.

For example, if the two points are (2, 3) and (7, 3), the horizontal distance between them is:

d = √((7 – 2)² + (3 – 3)²)

d = √(25)

d = 5

Therefore, the horizontal distance between the two points is 5.

In conclusion, the horizontal distance formula is a simple yet powerful tool that is used to calculate the distance between two points on a horizontal plane. It is based on the Pythagorean theorem and is widely used in many fields of study.

## Distance and Midpoint Formula

The distance formula is an essential tool in algebra that is used to find the distance between two points in a coordinate plane. It is commonly denoted as:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points. This formula is derived from the Pythagorean theorem and can be used to find the distance between any two points in a plane.

The midpoint formula is another useful tool that is used to find the midpoint between two points in a coordinate plane. It is given as:

$$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points. This formula can be used to find the midpoint of any line segment in a plane.

Both the distance and midpoint formulas are closely related and can be used to solve a variety of problems in algebra. For example, the distance formula can be used to find the length of a line segment, the distance between two points, or the radius of a circle. The midpoint formula can be used to find the center of a circle, the midpoint of a line segment, or the point that divides a line segment into two equal parts.

In addition, the distance and midpoint formulas can be used to solve a variety of equations in algebra. For example, the distance formula can be used to solve problems involving the Pythagorean theorem, while the midpoint formula can be used to solve problems involving the midpoint of a line segment.

Overall, the distance and midpoint formulas are essential tools in algebra that are used to solve a variety of problems involving points, lines, and circles in a coordinate plane.

## Using the Distance Formula FAQ

### How do you calculate distance between two points on a graph?

To calculate the distance between two points on a graph, you can use the distance formula which is derived from the Pythagorean theorem. The distance formula is as follows:

d = √((x2-x1)² + (y2-y1)²)

Where (x1, y1) and (x2, y2) are the coordinates of the two points.

### What is the distance formula in 3D space?

The distance formula in 3D space is similar to the distance formula in 2D space, except it includes a third dimension. The formula is as follows:

d = √((x2-x1)² + (y2-y1)² + (z2-z1)²)

Where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.

### What is the distance formula for finding the distance between two points in a plane?

The distance formula for finding the distance between two points in a plane is the same as the distance formula for finding the distance between two points on a graph. The formula is as follows:

d = √((x2-x1)² + (y2-y1)²)

Where (x1, y1) and (x2, y2) are the coordinates of the two points.

### How do you use the distance formula to find the length of a line segment?

To find the length of a line segment, you can use the distance formula. Simply plug in the coordinates of the endpoints of the line segment into the distance formula and solve for d. The value of d will give you the length of the line segment.

### What is the distance formula for finding the distance between two points in polar coordinates?

The distance formula for finding the distance between two points in polar coordinates is as follows:

d = √(r1² + r2² – 2r1r2cos(θ2-θ1))

Where (r1, θ1) and (r2, θ2) are the polar coordinates of the two points.

### What is the distance formula for finding the distance between two points on a sphere?

The distance formula for finding the distance between two points on a sphere is known as the great-circle distance formula. The formula is as follows:

d = r * arccos(sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(long2-long1))

Where r is the radius of the sphere, lat1 and lat2 are the latitudes of the two points, and long1 and long2 are the longitudes of the two points.

### What are two formulas for distance?

Two formulas for distance are the distance formula in 2D space and the distance formula in 3D space.

### What are the three formulas of distance?

There are three formulas of distance: the distance formula in 2D space, the distance formula in 3D space, and the great-circle distance formula for finding the distance between two points on a sphere.