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Pythagoras Theorem

Distance Between Two Points

We use the Pythagoras Theorem to derive a formula for finding the distance between two points in 2- and 3- dimensional space.

Let P = (x 1, y 1) and Q = (x 2, y 2) be two points on the Cartesian plane (see picture below).

                    So the horizontal distance between P and Q is x2 - x1 and the vertical distance between the points is y2 - y1.

Then from the Pythagoras Theorem we find that the distance between P and Q is

                    the square root of the horizontal distance between the points squared plus the vertical distance between the points squared.


In a similar way, it can be proved that if P = (x 1, y 1, z1) and Q = (x 2, y 2, z2) are two points in the 3-dimensional space, the distance between P and Q is


the square root of the sum of the squares of each of the x distance between the points, the y distance between the points and the z distance between the points.

Exercise 1.

Find the distance between two given two points in 2-D :

m= ( , )     n= ( , )


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