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Lesson One. Distance.



Distance:
The distance between two points ina plane maybe considerd in three cases.

Case 1!
P1(x1, y1) and P2(x2, y2) are two distinct points on the same horizontal line. The directed distance from P1 to P2 is:

P1P2=x2-x1


The undirected distance between P1 and P2 is:

|P1P2| = |x2 - x1| = |x1 - x2|


Case 2!!
P1(x1, y1) and P2(x1, y2) are two distinct points on the same vertical line. The directed distance from P1 to P2 is:

P1P2=y2-y1


The undirected distance between P1 and P2 is:

|P1P2| = |y2 - y1| = |y1 - y2|


Case 3!!!
P1(x1, y1) and P2(x1, y2) are two distinct points not on the same vertical or horizontal line. If a dine is drawn through P1 parallel to the x axis and another line is drawn through P2 parallel to the y axis, these lines will meet at point L(x2, y1), forming the right triangle P1LP2. Then |P1P2| is the measure |x2 - x1 and |y2 - y1, respectively. Therefore:

(P1P2)2 = (x2 - x1)2 + (y2 - y1)2


If d represents the undirected distance between the two points P1 and P2, then:

d = (square root)(x2 - x1)2 + (y2 - y1)2(end square root). This is equation (1-1)


EXAMPLE!!
The distance between A(-3,6) and B(3,-2) is calulated by equation (1-1) as

d=(square root) {3 - (-3)}2 + {-2-6}2(end square root)
d=(square root)62 + (-8)2(end square root)
d=10


Yay!




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