**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.

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 the point is 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 .

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?

coordinate is found by drawing a line through perpendicular to -axis, and coordinate is found by drawing a line through perpendicular to -axi. The vertical line intersects the -axis in , and horizontal line -axis in which means that the -coordinate of is , and -coordinate of is . This is writen as .

## The midpoint formula

How does one calculate the exact middle of the segment between and , if for example and .

Midpoint of a segment is halfway between the two end points of the segment. Midpoint will also have a and 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 of the first point, and and of 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, if and , then 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 two points and are two points:

It doesn’t matter which point is used as and which as .

For example, and .

## Parallel lines in the coordinate plane

Every line has a slope, which is a number that describes steepness of the line, or how much it angles from -axis. It is usually denoted by ““.

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

A slope is calculated by the formula:

Where and are any two points on that line.

Example 1:

If 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 and .

Example 3:

If the first line contains points and , and second line contains points and , are those lines parallel?

Their slopes are equal from that follows that the lines are parallel.

## Coordinate plane worksheets

Parallel lines and transversals

**Name the relationship** (647.5 KiB, 554 hits)

**Find the value of x** (809.9 KiB, 671 hits)

**Find the missing angle** (767.1 KiB, 546 hits)

**Find the angle and missing value** (840.6 KiB, 622 hits)

**Find the value knowing 2 parameters** (755.1 KiB, 451 hits)

Points on the coordinate plane

**Find the quadrant** (2.4 MiB, 539 hits)

**Plote points** (2.5 MiB, 514 hits)

**Find the coordinates of each point** (2.1 MiB, 515 hits)

The midpoint formula

**Integers** (207.6 KiB, 594 hits)

**Decimals** (211.6 KiB, 566 hits)

**Fractions** (737.4 KiB, 526 hits)

The distance formula

**Integers** (167.0 KiB, 592 hits)

**Decimals** (151.8 KiB, 368 hits)

**Fractions** (272.2 KiB, 404 hits)

**Square roots** (437.9 KiB, 431 hits)

Slope

**Find a slope** (1.9 MiB, 597 hits)