Examples of Using the Midpoint Formula

If I just graph this, it's going to look like the answer is "yes"; it's going to look like the line they gave me passes through the exact middle of the segment they gave me. But I have to remember that, while a picture can suggest an answer (that is, while it can give me an idea of what is going on), only the algebra can give me the exactly correct answer.

So, to be sure of my answer, I'll need to find the actual midpoint, and then see if the midpoint is actually a point on the line that they've proposed might pass through that midpoint.

First, I'll apply the Midpoint Formula to the segment's endpoints:

Now I'll check to see if this point is actually on the line whose equation they gave me. I'll take the equation, plug in the x-value from the midpoint (that is, I'll plug 3.2 in for x), and see if I get the required y-value of 1.4.

So, plugging the midpoint's x-value into the line equation they gave me did *not* return the y-value from the midpoint. While this line is very close to being a bisector (that is, while te line crosses the segment almost at the segment's middle point, as a picture would indicate), the line is not exactly a bisector (as the algebra proves).

So my answer is:

Yes, this exercise uses the same endpoints as did the previous exercise. But this time, instead of hoping that the given line is a bisector (perpendicular or otherwise), I will be finding the actual perpendicular bisector.

(Note: There are infinitely-many bisecting lines for the segment, but there is only one perpendicular bisector of the segment.)

This multi-part problem is actually typical of exercises you will probably encounter at some point when you're learning about straight lines. This is an example of a question where you'll be expected to remember the Midpoint Formula from however long ago you last saw it in class. Don't be surprised if you see this kind of question on a test in later math courses. Here's how to answer it:

First, I need to find the midpoint, since any bisector, perpendicular or otherwise, must pass through the midpoint. I'll apply the Midpoint Formula:

Now I need to find the slope of the line segment. I need this slope value in order to find the perpendicular slope for the line that will be the segment bisector.

I'll apply the Slope Formula:

The perpendicular slope (for my perpendicular bisector) is the negative reciprocal of the slope of the line segment. Remember that "negative reciprocal" means "flip it, and change the sign". So the slope of the perpendicular bisector will be:

With the perpendicular slope and a point (the midpoint, in this case), I can find the equation of the line that is the perpendicular bisector:

This line equation is what they're asking for. So my answer is:

Since the center of the circle is at the midpoint of any of the circle's diameter lines, I need to find the midpoint of the two given endpoints. I will plug the endpoints into the Midpoint Formula, and simplify:

This point is what they're looking for, but I need to specify what this point is. According to the exercise statement and what I remember from geometry, this midpoint is the center of the circle. So my answer is:

To find the coordinates of the other endpoint, I'm going to call those coordinates x and y, and then I'll plug these coordinates into the Midpoint Formula, and see where this leads.

I can set the coordinate expressions from the Formula equal to the given values, and then solve for the values of my variables.

Okay; that's one coordinate found. Now I'll do the other one:

Now that I've found the other endpoint coordinate, I can give my answer: