Monday, November 15, 2010

The Midpoint Formula - How to Determine the Midpoint of a Line Segment

If you want to know how to find the midpoint, the midpoint formula is a simple formula that will help you to determine the midpoint between two points on a graph.  Alternately, another use of the midpoint formula is to tell you the center point of a line segment, the point where the distance to one end of the line segment equals the distance to the other end of the line segment.  (In the end, you can prove your answer using the distance formula.)

So, how to find the midpoint between two points?  To determine the midpoint, you first need to know the x and y coordinates of your two points.  Two x coordinates and two y coordinates.  When you have these 4 values, the process is easy.

Basically, you look at the x values, and then you look at the y values.  (Make sure you don't mix them up!)  Once you have identified these values, all you have to do is take the average of each set.  Here is the general formula you will need to know:

Average x = (x1 + x2) / 2
Average y = (y1 + y2) / 2

Simple, right?  "Determine the midpoint between two lines" may sound difficult, but right there is how simple it can be!  Of course, since the midpoint will be a coordinate pair (both an x and a y), you would likely be expected to present your midpoint as a coordinate pair.  So, a more general way of representing this formula is:

(x1 + x2)(y1 + y2) )
        2               2

We're basically saying "we want the x-value that is halfway between these 2 x-values, and the y-value that is halfway between these 2 y-values."  And that is your midpoint coordinate.  Let's try a couple of examples.

Easy one to start.

Find the midpoint between (1,1) and (3,3).  So, how to find the midpoint?  Graph this out, and you will immediately be able to tell that the midpoint is going to be (2,2) just by looking at it, but let's apply our formula from above.

Average x = (x1 + x2) / 2 = (1+3) / 2 = 2
Average y = (y1 + y2) / 2 = (1+3) / 2 = 2
Midpoint (2,2)

So, our formula gave us what we already knew.  That's great, but let's try one that is a bit harder to see automatically.

Find the midpoint between (3,1) and (8,7).  (Remember to keep the x's together and y's together!  Don't put an x with a y, or it will be wrong!)  How to find the midpoint?

Average x = (x1 + x2) / 2 = (3+8) / 2 = 11/2 = 5.5
Average y = (y1 + y2) / 2 = (1+7) / 2 = 8/2 = 4
Midpoint (5.5, 4)

So, that was a bit tougher, but still pretty simple and we got the answer easily.  Let's try one more, a bit more complicated.

Find the midpoint between (-5,-2) and (-3, 10).  Same formula to solve it, but using negatives this time.

Average x = (x1 + x2) / 2 = ((-5)+(-3)) / 2 = (-5 - 3) / 2 = -8/2 = -4
Average y = (y1 + y2) / 2 = ((-2)+10) / 2 = (-2 + 10) / 2 = 8/2 = 4
Midpoint (-4,4)

So, there you go.  Now you know how to find the midpoint!  Hopefully this will help you now to determine the location of the midpoint on your line segments!  For an extra challenge/proof, use the distance formula to find the distance between your midpoint and each end of the line segment.  If the distances are the same, then you are obviously correct!

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