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Question 767747: How are the distance formula, the Pythagorean theorem, and equations of circles all related? How does one help you understand the others?
Answer by josgarithmetic(39620)   (Show Source):
You can put this solution on YOUR website! Too complicated to explain in a solution response posting here, but here is a try.
A little bit of number properties from "algebra" and some intuition with a couple of drawings for simpler geometry can lead you into a proof of the Pythagorean theorem. There is a square and area figure for this, or there is the President Garfield drawing of the trapezoid figure.
Later, you can look at the two dimensional formula of Pythagorean theorem, assume you have a hypotenuse of 1, and legs x and y. The theorem tells you that  , directly from the theorem. You had previously been often labeling legs a and b, and hypotenuse c, and saying  ; but now you can choose other variables, x and y, and assume c=1. You may then make a variable table for the x and y values of , and plot them. You do this on th x y coordinate system, the x and y axes. The form produced is a circle. In fact, this particular circle has radius 1 unit, is called the "unit circle", and leads well into the development of Trigonometry.
Distance Formula? This comes directly from the Pythagorean theorem, applied to a cartesian coordinate system. Plot two points. You can form a triangle between the two points. You can determine the legs's sizes using the coordinates of the points. Pythagorean theorem is then used to find the hypotenuse, which IS the distance from one point to the other point.
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