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Lesson by
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Mike McGarry
**

Magoosh Expert

Magoosh Expert

Coordinate Geometry. One of the most elegant ideas in all of mathematics, is the idea of the coordinate plane. Other names include the x-y plane, the rectangular coordinate plane, and the Cartesian plane. And that final name is in honor of the person who discovered it, the French mathematician Rene' Descartes.

Descartes' brilliant idea began, by simply putting two number lines at right angles to each other. So of course we know a number line has whole numbers on it, it has fractions, it has decimals, and it goes on forever in both the positive direction and the negative direction. And so what we have here really, are just two number lines crossing.

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. And of course each one of them goes on forever. Each one of them contains positive whole numbers, negative whole numbers. Positive fractions and decimals, negative fractions and decimals. The whole nine yards.

The point where the axis cross, zero on each axis, is called the origin, and that's considered the center of the entire plane. Of course, this allows us to indicate the position of any point in the plane, by the x and y-coordinate of the point. So for example, we look at this particular point. This particular point, is vertically above x = 5.

So the x-coordinate has to be 5. It is on the same horizontal line, as y = 4. So, its y-coordinate is 4. And its position is given by (5, 4). That is the ordered pair that denotes the exact position of that point. As you may remember, (5, 4) is an ordered pair, with an x-coordinate followed by a y-coordinate.

So they're in alphabetical order, first the x-coordinate, then the y-coordinate. Every one of the infinite number of points in the plane, can be indicated by a unique ordered pair. So that's amazing fact number one. You could go to any position in the plane, an infinite number of points in the plane, every single one will have a unique ordered pair, a unique x-y coordinate denoting it's exact location.

On the test, given an ordered pair, you need to be able to locate that point and give it a point. Picture of a point, you need to be able to figure out, what the coordinates for that point are, so that is an absolutely essential skill. Here's a very simple practice problem. Pause the video and then we'll talk about this.

Okay. So this is actually much easier than anything you'll see on the test. It may be that this would be part of another problem on the test. But, we want to know, what are the coordinates of this point. Well. First of all notice that we're to the left of the y-axis.

We're on the left side of the x-y plane. And so this would be where that horizontal number line is negative. And so the x-axis, because we're to the left of zero, we're in the negative part of that axis. So this is gonna have a negative x-coordinate. So we count backwards, one, two, three, four, five, six, seven and then we count up one two, three, four.

So that means that the x-coordinate is negative 7, the y-coordinate is positive 4, and the coordinates of that point are negative 7, 4. That is the unique ordered pair, which gives the exact location of that point. The axis divide the entire plane into four regions, known as quadrants. These quadrants are denoted, clockwise from the upper right, as I, II, III, and IV.

And they're almost always denoted with four Roman numerals like this. If we know the quadrant of a point, we immediately know the positive or negative sign of both the x-coordinate and the y-coordinate. So for example, in the first quadrant, both coordinates are positive. In the second quadrant The x's are negative but the y's are positive. In the third quadrant, both the coordinates are negative at that point.

Everything is negative in the third quadrant. In the fourth quadrant the x's are positive but the y's are negative. It's also important to note that any point that is exactly on the x-axis or exactly on the y-axis or certainly the origin, these are not in any of the four quadrants. So the four quadrants are only for points that are off the axis.

Here's a practice problem. Pause the video and then we'll talk about this. Okay, this is a problem that actually could appear on the test. Because it's a little less straightforward and requires a little bit of visualization. Point M is the midpoint of segment AB.

If A = 2 common negative 3, and M is on the negative x-axis, in what quadrant is B? So let's visualize this. We have A here. We don't know where M is, but M is going to be on the negative x-axis. The negative x-axis here.

So let's just pick a random point, we could even pick one relatively close to the origin, right there. So if A goes to M, well then B would have to be up here. And it turns out, that no matter where we put M on that axis. We could move it back and forth. B is always gonna wind up in the second quadrant.

So really, the answer to this question is quadrant number 2. In summary. You need to know the terms. Origin, x-axis, y-axis, x-coordinate, and y-coordinate. Those are terms that tests will use, and you need to be able to recognize them and know what they mean.

It's important to appreciate that every single point in the plane, infinite number of points in the plane, every single one can be noted by a unique ordered pair. A unique set of x and y-coordinates. And finally, the axis divide the plane into four quadrants. The quadrants of a point determines the positive and negative signs of its x and y-coordinates, and the test likes to ask about quadrants.

Show TranscriptDescartes' brilliant idea began, by simply putting two number lines at right angles to each other. So of course we know a number line has whole numbers on it, it has fractions, it has decimals, and it goes on forever in both the positive direction and the negative direction. And so what we have here really, are just two number lines crossing.

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. And of course each one of them goes on forever. Each one of them contains positive whole numbers, negative whole numbers. Positive fractions and decimals, negative fractions and decimals. The whole nine yards.

The point where the axis cross, zero on each axis, is called the origin, and that's considered the center of the entire plane. Of course, this allows us to indicate the position of any point in the plane, by the x and y-coordinate of the point. So for example, we look at this particular point. This particular point, is vertically above x = 5.

So the x-coordinate has to be 5. It is on the same horizontal line, as y = 4. So, its y-coordinate is 4. And its position is given by (5, 4). That is the ordered pair that denotes the exact position of that point. As you may remember, (5, 4) is an ordered pair, with an x-coordinate followed by a y-coordinate.

So they're in alphabetical order, first the x-coordinate, then the y-coordinate. Every one of the infinite number of points in the plane, can be indicated by a unique ordered pair. So that's amazing fact number one. You could go to any position in the plane, an infinite number of points in the plane, every single one will have a unique ordered pair, a unique x-y coordinate denoting it's exact location.

On the test, given an ordered pair, you need to be able to locate that point and give it a point. Picture of a point, you need to be able to figure out, what the coordinates for that point are, so that is an absolutely essential skill. Here's a very simple practice problem. Pause the video and then we'll talk about this.

Okay. So this is actually much easier than anything you'll see on the test. It may be that this would be part of another problem on the test. But, we want to know, what are the coordinates of this point. Well. First of all notice that we're to the left of the y-axis.

We're on the left side of the x-y plane. And so this would be where that horizontal number line is negative. And so the x-axis, because we're to the left of zero, we're in the negative part of that axis. So this is gonna have a negative x-coordinate. So we count backwards, one, two, three, four, five, six, seven and then we count up one two, three, four.

So that means that the x-coordinate is negative 7, the y-coordinate is positive 4, and the coordinates of that point are negative 7, 4. That is the unique ordered pair, which gives the exact location of that point. The axis divide the entire plane into four regions, known as quadrants. These quadrants are denoted, clockwise from the upper right, as I, II, III, and IV.

And they're almost always denoted with four Roman numerals like this. If we know the quadrant of a point, we immediately know the positive or negative sign of both the x-coordinate and the y-coordinate. So for example, in the first quadrant, both coordinates are positive. In the second quadrant The x's are negative but the y's are positive. In the third quadrant, both the coordinates are negative at that point.

Everything is negative in the third quadrant. In the fourth quadrant the x's are positive but the y's are negative. It's also important to note that any point that is exactly on the x-axis or exactly on the y-axis or certainly the origin, these are not in any of the four quadrants. So the four quadrants are only for points that are off the axis.

Here's a practice problem. Pause the video and then we'll talk about this. Okay, this is a problem that actually could appear on the test. Because it's a little less straightforward and requires a little bit of visualization. Point M is the midpoint of segment AB.

If A = 2 common negative 3, and M is on the negative x-axis, in what quadrant is B? So let's visualize this. We have A here. We don't know where M is, but M is going to be on the negative x-axis. The negative x-axis here.

So let's just pick a random point, we could even pick one relatively close to the origin, right there. So if A goes to M, well then B would have to be up here. And it turns out, that no matter where we put M on that axis. We could move it back and forth. B is always gonna wind up in the second quadrant.

So really, the answer to this question is quadrant number 2. In summary. You need to know the terms. Origin, x-axis, y-axis, x-coordinate, and y-coordinate. Those are terms that tests will use, and you need to be able to recognize them and know what they mean.

It's important to appreciate that every single point in the plane, infinite number of points in the plane, every single one can be noted by a unique ordered pair. A unique set of x and y-coordinates. And finally, the axis divide the plane into four quadrants. The quadrants of a point determines the positive and negative signs of its x and y-coordinates, and the test likes to ask about quadrants.