Geometry: Coordinate Geometry

On this page we hope to clear up any problems that you might have with coordinte geometry. Read on or follow any of the links below to start understanding coordinate geometry better!

Midpoint formula
Slope of lines
Equations of lines
Distance formula
Quiz on coordinate geometry

Midpoint Formula

A midpoint is a point that denotes the middle of any given line segment. The Midpoint Theorem says the x coordinate of the midpoint is the average of the x coordinates of the endpoints and the y coordinate is the average of the y coordinates of the endpoints.

If a line segment has the end points (x1, y2) and (x2, y1), the midpoint is given by the following formula (the variables would be subscripted if text-only browsers allowed subscripted text): [(((x1) + (x2))/2), (((y1) + (y2))/2)].

Example:

1.  Problem: Find the coordinates (x, y) of the midpoint of the
             segment that connects the points (-4, 6) and (3, -8).
   Solution:     x1 + x2   -4 + 3   -1     1
             x = ------- = ------ = -- = - -
                    2         2      2     2

                 y1 + y2   6 + (-8)   -2
             y = ------- = -------- = -- = -1
                    2          2       2

             The answer: (-.5, -1)

Slope of a Line

Finding the slope of a line is a topic usually covered in Algebra I (Elementary Algebra) courses. We followed this custom on our site. You can follow this link to go to a page that describes the process of finding the equation of a line (finding the slope is a main step in this process).

Equations of Lines

Finding the equation of a given line is usually covered in Algebra I (Elementary Algebra) courses. This custom was followed on this site. You can follow this link to go learn about finding the equation of a line.

Although finding the equation of a line is covered, you sometimes get more complex problems where you are given some information about a graph and are expected to find the equation of the line described.

Example:

1.  Problem: Write the equation of the line that passes through
             the point (2, -3) and has a slope of (1/2).  Use
             slope-intercept form.
   Solution: Write the general equation used for the slope-
             intercept form.

             y = mx + b

             Plug in any given information.

             -3 = (1/2)2 + b

             -3 = 1 + b
             -4 = b

             Write the equation of the line in slope-intercept form:
             y = .5x - 4

Distance Formula

The distance formula says that the distance d between any two points with coordinated (x1, y1) and (x2, y2) is given by the following equation (the variables would be subscripted if text-only browsers allowed for subscripted text): d = SQRT[((x2) - (x1))^2 + ((y2) - (y1))^2].

Example:

1.  Problem: Find the distance between (-2, 3) and (8, -1).
   Solution: Plug any given information into the distance
             equation.

             d = SQRT[(8 - (-2))^2 + (-1 - 3)^2]

             Simplify.

             d = SQRT[10^2 + (-4)^2]
             d = SQRT(100 + 16)
             d = 2(SQRT(29))

Take the quiz on coordinate geometry. The quiz is very useful for either review or to see if you've really got the topic down.

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