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Cartesian Coordinate System,
Distance and Midpoint Formula

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Cartesian Coordinate System, Distance and Midpoint Formula

The Cartesian Coordinate System consists of two number lines perpendicular to each other at their 0's. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. This allows us to assign to each point in a plane a coordinate. Each coordinate consists of a pair of numbers, the first of which is the x-coordinate and the second is the y-coordinate, written (x,y). For example, the point A (2,-3) is a point 2 units to the right of the origin (the point (0,0)) and then 3 units down. And point B (-4,1) is a point 4 units to the left of the origin and then 1 unit up.

We can find the distance between any two points in the xy-plane by using a revised version of the Pythagorean Theorem called the distance formula.

where d is the distance between the two points (x_a,y_a) and (x_b,y_b).

Example 1: Find the distance between the points A and B given above.

Note: the order of the points A and B does not make a difference (try the above problem with the points switched).

We can also find the midpoint between these two points by taking the average of their coordinates. This translates into the following formula:

midpoint =

Example 2: Find the midpoint between the points A and B given above.

midpoint =

Example 3: Find the distance between point A (-4,-2) and the midpoint of the segment joining points B (1,5) and C (-3,3).

First we need to find the midpoint of segment BC.

Now we need to find the distance between this point
and point A.