Proof Essay

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PROOF OF THE DISTANCE FORMULAS First Proof: Distance between two points co ordinates is a basic concept in geometry.Now, we give an algebraic expression for the same. Let P1 (x1, y1) and P2 (x2, y2) be two points in a Cartesian plane and denotes the distance between P1 and P2 by d(P1, P2) or by P1P2. Draw the line segment The segment is parallel to the x axis Then y1 = y2. Draw P1 L and P2 M, perpendicular to the x-axis. Then d(P1,P2) is equal to the distance between L and M. But L is (x1, 0) and M is (x2, 0). So the length LM = |x1-x2| Hence d (P1, P2) = |x1-x2|. therefore, [d(P1,P2)]2= |x1-x2|2+ |y1-y2|2 =(x1-x2)2+(y1-y2)2 =(x2-x1)2+(y2-y1)2 d(P1,P2) = Second Proof The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry. Here's how we get from the one to the other: Suppose you're given the two points (–2, 1) and (1, 5), and they want you to find out how far apart they are. The points look like this: | | | You can draw in the lines that form a right-angled triangle, using these points as two of the corners: | | | It's easy to find the lengths of the horizontal and vertical sides of the right triangle: just subtract the x-values and the y-values: | | | Then use the Pythagorean Theorem to find the length of the third side (which is the hypotenuse of the right triangle): c2 = a2 + b2

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