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Published byAlejandro Fowler Modified over 2 years ago

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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: a 2 + b 2 = c = c = c = c 2 so that c 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 generalize the points A and B to be at any coordinate, for example A can be at (x 1,y 1 ) and B can be (x 2,y 2 ). Since the points are identified in generic terms, we can see the legs of the right angle as y 2 – y 1 for the vertical length and x 2 – x 1 for the horizontal length.

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Now, using the Pythagorean Theorem, (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = (distance) 2 solve for distance by taking the square root of both sides…

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distance = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 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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