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**Proving the Distance Formula**

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**What is the distance between points A and B?**

We can use the Pythagorean Theorem to find the distance.

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**The point A is at (6, 4) and the point B is at (-3, -4).**

Drawing lines to create a right triangle gives us:

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With the creation of the right triangle, the length of the hypotenuse (AB) is the distance from point A to point B. The length of AC is the difference in the y-coordinates of the points, or 4 – (-4), which is 8. The length of BC is the difference in the x-coordinates of the points, or 6 – (-3), or 9. Use the Pythagorean Theorem to find the length of AB!!

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**Using the Pythagorean Theorem, we get:**

a2 + b2 = c = c = c2 145 = c2 so that c ≈ 12.04 So, the length of AB is 12.04, meaning that the distance from A to B is about units.

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**We can now generalize this process for any two points**

We can now generalize this process for any two points. We can generalize the points A and B to be at any coordinate, for example A can be at (x1,y1) and B can be (x2,y2). Since the points are identified in generic terms, we can see the legs of the right angle as y2 – y1 for the vertical length and x2 – x1 for the horizontal length.

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**Now, using the Pythagorean Theorem, **

(x2 – x1)2 + (y2 – y1)2 = (distance)2 solve for distance by taking the square root of both sides…

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

You have just derived the distance formula for any two points in the coordinate plane! (Use this formula to plug in Point A, (3,4), and Point B, (-3,-4), and see if your distance is the same as it was when we used the Pythagorean Theorem.)

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Not only can we use the distance formula to find the distance between Point A and Point B, we can also find the slope of the line using our two points. This indicates that the direction between these two points follows a line with a slope of 8/9.

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