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# coord_geom - The distance formula and the mid-point formula...

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The distance formula and the mid-point 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 right-angled 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 .

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coord_geom - The distance formula and the mid-point formula...

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