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Coordinates in Four Quadrants

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You’ve probably come across coordinates in some form before. Maybe you’ve seen a map, and been asked to locate the position of something, or maybe you’ve had practice reading from a line graph and not even realised you were using coordinates. And what did those coordinate systems look like? Most likely they looked like this, a number line counting from zero upwards, and a number line counting from zero going out to the right.

These number lines are known as **axes** and together, these two number lines create what is known as the **coordinate plane**. The coordinate plane is a useful tool, as we can describe any location (point) on it with a number on the horizontal axis, and a number on the vertical axis.

Well, the coordinate plane doesn’t stop there. In fact, this simple coordinate plane isn’t the whole story at all, you could even say it’s only a quarter of the story...

So what exactly do we mean when we talk about **coordinates in four quadrants?**

The simple coordinate plane you are used to seeing is only one of the four 2-dimensional coordinate **quadrants**. What do we mean by quadrants? Well, start by drawing the simple coordinate plane axes we are familiar with. It should look something like this.

Now, extend the vertical line downwards toward the bottom of the page. Done that? Good!

Now extend the horizontal line backwards, toward the left side of the page. You should be left with something like this.

So what’s happened to our simple coordinate system?

Well, by extending each line we have actually revealed more of the coordinate plane. We can see that the simple system consisting of two perpendicular positive number lines was actually just one of four separate areas created by our axes. Each of these four areas is called a **quadrant**, from the Latin *quadrans* meaning *four*.

From here, all we need to do to understand our four-quadrant coordinate plane is to label each of the axes with numerical values. If we keep zero in the same place on each axis, where the two axes cross, then it makes sense that to the left along the horizontal axis, we count down from zero: –1, –2, –3 ... and do the same going vertically downward from zero on the vertical axis: –1, –2, –3...

So what exactly do we mean by coordinates in four quadrants? Well...

**Coordinates in four quadrants** are coordinates plotted in a coordinate system in which both axes extend in both positive and negative directions creating four quadrants.

But how exactly do we graph coordinates in four quadrants? Let's take a look!

As in any two-axis coordinate plane, any point is made up of an -coordinate and -coordinate, in the form . In a four-quadrant coordinate system,

any point in the upper-right quadrant, the

**first quadrant**, will have both a positive -coordinate and positive -coordinate.

and any point in the upper-left quadrant, the

**second quadrant**, will have a negative -coordinate, and positive -coordinate.

On the other hand,

any point in the lower-left quadrant, the

**third quadrant**, will have both a negative -coordinate and negative -coordinate;

and any point in the lower-right quadrant, the

**fourth quadrant**, will have a positive -coordinate, and negative -coordinate.

**(1)**

Plot the point on a four-quadrant coordinate plane.

**Solution:**

The point has a positive -coordinate and negative -coordinate, therefore it will be in the lower-right quadrant.

**(2)**

Plot the point on a four-quadrant coordinate plane.

**Solution:**

The point has a negative -coordinate and positive -coordinate, therefore it will be in the upper-left quadrant.

**(3)**

Plot the point on a four-quadrant coordinate plane.

**Solution:**

The point has a positive -coordinate and positive -coordinate, therefore it will be in the upper-right quadrant.

**(4)**

Plot the point on a four-quadrant coordinate plane.

**Solution:**

The point has a negative -coordinate and negative -coordinate, therefore it will be in the lower-left quadrant.

There's more to coordinates in four quadrants than just plotting them on a four-quadrant coordinate plane. Let's take a look at some more examples.

**(1)**

What is the distance between points and ?

**Solution:**

Reading from the coordinate plane point has coordinates , and point has coordinates

We can find the distance between the two points by using Pythagoras' theorem, .

So

**(2)**

Is point or closer to the point ?

**Solution:**

Reading off the coordinate plane, point has the coordinates and has coordinates .

Firstly, we must find out how far is from the point .

And then we find out how far is from the

- The full coordinate plane is divided into four quadrants by the horizontal and vertical axes.
- The quadrant that any point is located within can be determined by the sign of each of that point's coordinates.

Coordinates in four quadrants are made up of an x-coordinate and y-coordinate of the form (x, y).

The x-coordinate is how far along the horizontal axis a point is, and the y-coordinate is how far along the vertical axis a point is.

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