# The Distance Formula: Finding the Distance Between Two Points

TLDRIn this educational video, Professor Dave delves into the mathematical journey of calculating the distance between two points on a coordinate plane, leveraging the foundational Pythagorean Theorem. By introducing a practical scenario, the video guides viewers through constructing a right triangle, using the coordinates of two points to determine the lengths of the triangle's legs. Applying the theorem, the distance formula is derived, simplifying the process to a straightforward equation. Through a clear example, the video demonstrates how this formula easily calculates the distance, reinforcing the concept with a hands-on exercise. This explanation demystifies the distance formula, making it accessible and understandable, showcasing its practical application in algebraic problems.

###### Takeaways

- ๐ The distance formula allows us to calculate the distance between two points on the coordinate plane.
- ๐ To use the distance formula, we construct a right triangle between the two points. The hypotenuse is the distance we want to find.
- ๐ The legs of the triangle are the differences between the x-coordinates and y-coordinates of the two points.
- ๐ค We can then use the Pythagorean Theorem to relate the legs and the hypotenuse.
- ๐ข The final distance formula squares the leg lengths, adds them, and takes the square root.
- ๐งฎ The formula uses the x- and y-coordinates directly, allowing us to easily calculate distance.
- ๐ To find the distance between (-2,2) and (2,5), plug the points into the formula to get 5.
- ๐ญ The distance formula works for any two points,without needing to draw the triangle.
- โ๏ธ Using the formula gives the same result as constructing a triangle and using the Pythagorean Theorem.
- ๐ง The distance formula allows fast, generalized distance calculations on the coordinate plane.

###### Q & A

### What formula does Professor Dave derive in the video?

-Professor Dave derives the distance formula, which allows you to calculate the distance between two points on the coordinate plane.

### What theorem does Professor Dave use to derive the distance formula?

-Professor Dave uses the Pythagorean Theorem to derive the distance formula.

### What are the legs and hypotenuse in the right triangle constructed by Professor Dave?

-The legs are the horizontal and vertical line segments connecting the two points. The hypotenuse is the line segment connecting the two points, which represents the distance.

### What do the variables in the formula represent?

-X1 and Y1 represent the x and y coordinates of the first point. X2 and Y2 represent the x and y coordinates of the second point. D represents the distance between the two points.

### Why does Professor Dave only take the positive square root at the end?

-Length cannot be negative, so only the positive square root, which gives a positive value for the distance, makes sense.

### What are the two points Professor Dave uses in the example?

-In the example, Professor Dave uses the points (-2, 2) and (2, 5).

### What is the distance calculated between those two points?

-Using the distance formula, the distance calculated between (-2, 2) and (2, 5) is 5.

### How can you check if the distance formula gives the right answer?

-You can construct a triangle between the two points and use the Pythagorean Theorem to calculate the distance. This should match the value from the distance formula.

### When would you need to use the distance formula?

-You would use the distance formula any time you have two points on a coordinate plane and need to find the distance between them.

### Can the distance formula be used in three-dimensional space?

-No, the distance formula only works for two-dimensional space, specifically on the standard xy-coordinate plane. A different formula is needed for three-dimensional space.

###### Outlines

##### ๐ Deriving the Distance Formula

Explains how to use the Pythagorean theorem to derive a formula for calculating the distance between two points (x1,y1) and (x2,y2) on the coordinate plane. Constructs a right triangle with the line segment between points as the hypotenuse and the differences in x and y values as the legs. Applies Pythagorean theorem: (x2 - x1)2 + (y2 - y1)2 = D2, where D is the distance. Taking the square root solves for D.

##### ๐ Applying the Distance Formula

Demonstrates using the derived distance formula to calculate the distance between the points (-2,2) and (2,5). Plugs the x and y values into the formula: โ(4)2 + (3)2 = โ16 + 9 = โ25 = 5. Checks this against constructing a triangle and using Pythagorean theorem directly.

###### Mindmap

###### Keywords

##### ๐กcoordinate plane

##### ๐กPythagorean Theorem

##### ๐กdistance formula

##### ๐กhypotenuse

##### ๐กhorizontal line segment

##### ๐กvertical line segment

##### ๐กlegs

##### ๐กcoordinates

##### ๐กline segment

##### ๐กlength

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###### Transcripts

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