# Coordinate plane – The midpoint and distance formula

Coordinate plane is a plane such that each point is uniquely specified by a pair of numerical coordinates. In the plane there are two vertical lines, called axis, the horizontal one is the -axis and  the vertical one is the -axis. Their intersection is called the point of origin.

A coordinate is an ordered pair which tells us the location of the point it represents. The first number in the ordered pair tells us how far is the point located from the point of origin on the -axis, and the other number how far on the -axis.

Both axis have a positive and a negative side. Negative side of the -axis is found on the left from the point of origin, and positive on the right. Negative side of the -axis is found down from the point of origin, and positive up.

The point of origin is always in .

The two axes divide the plane into four sections called quadrants. The quadrants are labelled with Roman numerals, starting at the positive -axis and going around anti-clockwise. All the points in the first quadrant have both coordinates positive, in the second quadrant all the coordinates are negative and coordinates positive, in the third both coordinates are negative, and in the fourth the  coordinates are positive and  coordinates negative.

For example, is a point in the first quadrnat, in the second quadrant, in the third quadrant and in the fourth.

How to determine coordinates of specific points located in a coordinate plane?

Let’s see it on an example where you have to find coordinates for point A. Numbers on the number line indicate how large your basic measure is. Every segment between two points that show you how far you are from origin point is of constant lenght.

Now, back to finding coordinates. Coordinates always come in form of ordered pair. So A will have coordinates .

X coordinate is found by drawing a perpendicular line from A to -axis. And y coordinate is found by drawing a horizontal line from A to -axis. Their intersection with axes will show you your coordinates.

The vertical line intersects the -axis in , and horizontal line -axis in . That means that -coordinate of will be , and -coordinate of will be . This is writen as .

## The midpoint formula

If we have two points in the coordinate plane, for example  and , what can interest some is how to calculate the exact middle of the segment between and .

The total distance the line crossed on individual axes can be counted. In our example, the total distance the line crossed on the -axis is  and on the -axis is . It is simply counting how many basic measures exist between and coordinates of the points.

Midpoint of a segment is halfway between the two end points of the segment. Midpoint will also have a x and y coordinate. Its coordinate is halfway between the coordinates of the end points, and its coordinate is halfway between the coordinates of the end points. To calculate it add modules of coordinates and divide by 2, and the same for coordinates.

In the previous example: , .

The exact formula is: where and are coordinates from the first point, and and from the second.

## The distance formula

The distance between points and is marked with a modul: . And represents the shortest lenght from one point to another. For example,  and .

The distance between and is easy to determine since their coordinates are the same so all you have to do is add length from the left to the one on the right. That means: . Similarly  and share the coordinate so: .

But what to do with the distance between and ? As you maybe have already noticed, triangle is a right angled triangle, and the distance between and is the same as the length of its hypotenuse. Now that we know that we can easily calculate our distance from the Pythagorean theorem.

So

However this is a longer way, so let discuss a shorter one.

If you have two points and :

And it does not matter which point you use as and which as .
Let’s try it out on for points and .

## Parallel lines in the coordinate plane

Every line has a slope. The slope of a line is a number that describes steepness of the line, or how much it angles from -axis.

If the slopes of two lines are equal, then the lines are parallel.

A slope is calculated by the formula:

Slope =

Where and are any two points on that line.

Example: Let’s say we know two points on our line: and . Calculate the slope.

Slope is usually marked with an ““.

The slope of this line equals to .

Example 2: Calculate the slope of the given line.

First you choose two points trough which the line goes trough. For instance, we’ll use (-2, 0) and (0, 1).

As we already said, if two lines are parallel, their slopes are the same.

Example:

If the first line contains points A(2, 4) and B(3, 2), and second line contains points C(4,8) and D(6, 4), are those lines parallel?

Their slopes are equal, which indicates that the lines are parallel.

## Coordinate plane worksheets

Parallel lines and transversals

Name the relationship (647.5 KiB, 508 hits)

Find the value of x (809.9 KiB, 613 hits)

Find the missing angle (767.1 KiB, 499 hits)

Find the angle and missing value (840.6 KiB, 571 hits)

Find the value knowing 2 parameters (755.1 KiB, 424 hits)

Points on the coordinate plane

Find the quadrant (2.4 MiB, 508 hits)

Plote points (2.5 MiB, 474 hits)

Find the coordinates of each point (2.1 MiB, 470 hits)

The midpoint formula

Integers (207.6 KiB, 556 hits)

Decimals (211.6 KiB, 541 hits)

Fractions (737.4 KiB, 502 hits)

The distance formula

Integers (167.0 KiB, 553 hits)

Decimals (151.8 KiB, 345 hits)

Fractions (272.2 KiB, 363 hits)

Square roots (437.9 KiB, 399 hits)

Slope

Find a slope (1.9 MiB, 551 hits)

Shares