دوره GRE Test- Practice & Study Guide ، فصل 24 : GRE Quantitative Reasoning- Coordinate Geometry

درباره‌ی این فصل:

Learn about coordinate geometry concepts covered on the GRE with this chapter's fun and engaging lessons. You can also test your knowledge with lesson quizzes and chapter exams to make sure your ready to tackle the math section of the GRE.

Line Segments and Midpoints

This lesson is on the midpoint formula. Now, in order to understand what the formula is, it’s probably best to know what a midpoint itself is. But, in order to know what a midpoint is, we have to know what a line segment is. Luckily, neither of these things is too complicated, so it should take only a minute to go over them.

First off, a line segment : as the name implies, it’s pretty much a line, but it’s only one piece of a line. A real line goes on forever in both directions. But, a line segment has two endpoints; it kind of stops on either end.

Because they don’t go on forever, every line segment has a spot that is right in the middle - the midpoint . Whether the line segment is long, or short, or horizontal or any other way, the midpoint is the point right in the middle. Knowing exactly where the midpoint is can be handy for a number of reasons, but it isn’t always obvious how to find it.

When the line segment is either perfectly horizontal or perfectly vertical, the only things you need to know how to do is count and then divide by 2. But, when the line segment is diagonal (like below), knowing exactly where the middle is isn’t quite so straightforward.

Finding the midpoint of a diagonal line is not straightforward. diagonal line

The Midpoint Formula

So, what do you do? Well that’s what the midpoint formula is made for. As long as we have the endpoints, we can simply substitute the values in, and the midpoint coordinates pop out!

Applying the midpoint formula to a diagonal line applying midpoint formula

But, formulas like this are easy to forget, so let’s quickly take a look at where this comes from. That way, in case you do forget, you can still figure it out the long way. All the formula really does is find the middle, or the average, of the x s and then the middle, or the average, of the y s, separately.

In our previous example, the line segment started at x = 4 and ended up at x = 10. You could just count your way to the middle, or find the average by adding them together and then dividing by 2. Either way, we find that the midpoint must be at where x = 7.

The y s, on the other hand, went from 1 to 5. That makes the y -coordinate the number right in the middle, which is 3. And, sure enough, there’s our midpoint, (7, 3)!

Example 1

Throwing some negative numbers into the mix can make it a little trickier, so let’s try a few more for some quick practice. How about this: find the midpoint between the two points (-5, 3) and (-1, -3).

We could use the formula by calling the first x , x 1, the second one, x 2. Do the same thing with the y s, and substitute the values in like below:

Example 1 example 1 for midpoint formula

The x s are pretty straightforward, but notice how the y s have canceled out and turned into a 0? Nothing wrong with that! Zero divided by anything is just zero, so that makes our midpoint (-3, 0).

Example 2

Last example: find the midpoint between (1, 6) and (2, -2). When we substitute into our formula, we notice that this example isn’t going to give us as nice of an answer as the rest of them did. That’s because we end up with an odd number when we add the x s together. And, sure enough, halfway between 1 and 2 is 1.5!

Example 2 example 2

Lesson Summary

To quickly review what we’ve learned:

The midpoint formula helps us find the point that is in the exact middle of any line segment.

If we know the endpoints of the line segment , then the midpoint has the coordinates given by this formula:

The midpoint forumla midpoint formula

All this is really is the average, or the middle, of the x -coordinates, and the average of the y s as well.

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