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Unformatted text preview: The distance formula and the midpoint formula The distance formula . Given two points A ( ) , x 1 y 1 and B ( ) , x 2 y 2 in the coordinate plane where, in the first instance, B is above and to the right of A as in the picture, the horizontal distance between the two points is  x 2 x 1 and the vertical distance between the two points is  y 2 y 1 . If the distance between the two points A and B is d , applying Pythagoras' theorem in the rightangled triangle ABC , where AC is parallel to the x axis and BC is parallel to the y axis, it follows that = d 2 + ( ) x 2 x 1 2 ( ) y 2 y 1 2 . Similar diagrams can be drawn for the other possible configurations for A and B . If B is to the right and below A , then the horizontal distance between A and B is still  x 2 x 1 but the vertical distance is  y 1 y 2 . Because = ( ) y 1 y 2 2 ( ) y 2 y 1 2 , applying Pythagoras' theorem in the triangle ABC still gives = d 2 + ( ) x 2 x 1 2 ( ) y 2 y 1 2 . The other two possible configurations with B to the left of A also give the same result. Hence in all cases the distance d between A and B is given by = d + ( ) x 2 x 1 2 ( ) y 2 y 1 2 . ____________ Example 1 : The distance d between the points ( ) , 1 3 and ( ) , 5 6 is: = d + ( ) 5 ( ) 1 2 ( )  6 ( ) 3 2 = + 6 2 ( ) 3 2 = + 36 9 = 45 = 9 . 5 = 3 5....
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This note was uploaded on 11/12/2011 for the course MATH 111 taught by Professor Uri during the Spring '08 term at Rutgers.
 Spring '08
 Uri
 Distance Formula

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