The Length of a Line
The distance between two points on a coordinate plane is the shortest route
between the two points. Unless the two points have the same x- or y-coor-
dinate, the distance between them will be represented by a diagonal line.
This diagonal line is always the hypotenuse of a right triangle that exists
(but is not drawn) in the coordinate plane. (One of the legs of this imagi-
nary triangle is a horizontal line, parallel to the x-axis and the other leg is
a vertical line, parallel to the y-axis.) The length of the diagonal
hypotenuse is the square root of the sum of the squares of the two legs. This
is the Pythagorean theorem in reverse. (Note! In the following equations,
the subscript numbers are used to distinguish the points from each other:
(x 1 , y 1 ) means point one, and (x 2 , y 2 ) means point two.)
The distance, d, between any two points A(x 1 , y 1 ) and B(x 2 , y 2 ) in the
coordinate plane is d= (x 2 – x 1 )^2 + (y 2 – y 1 )^2
c^2 = a^2 + b^2 (Pythagorean theorem)
a= horizontal leg: (x 2 – x 1 )
b= vertical leg: (y 2 – y 1 )
c= d(the distance between two points)
d= (x 2 – x 1 )^2 + (y 2 – y 1 )^2
501 Geometry Questions