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 is in honor of the person who discovered it, the French mathematician Rene Descartes. Show Transcript
Descartes's brilliant idea began by simply putting two numbers 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 axes 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 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 on 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 its exact location.
On the test, given an ordered pair, you need to be able to locate that point and give it a point, you need a picture of a point. You need to be able to figure out what the coordinates for that part 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.
So this is actually much easier than you'll see on the test. It maybe that this would be part of another problem on the test, but we wanna know what are the coordinates of this point. Well, first of all, notice that we are to the left of the x-axis. We are 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 the x-coordinate is -7, the y-coordinate is positive 4 and the coordinates to that points are -7, 4.
That is the unique ordered pair, which gives the exact location of that point. The axes 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 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 xs are negative, but the ys are positive. In the third quadrant, both the ordinates are negative at that point. Everything is negative in the third quadrant. In the fourth quadrant, the xas are positive, but the ys are negative.
So 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 axes. Here's a practice problem. Pause the video and then we'll talk about this.
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, -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 is here. So let's just pick a random point, we need to 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 no matter where we put M on that axis, we can 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 II. In summary, you need to know the terms. Origin, x-axis, y-axis, x-coordinate and y-coordinate. Those are terms the test 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.