Coordinate Geometry

In this lesson, we will learn

the coordinate plane or Cartesian plane
the slope formula
the equation of a line
the slopes of parallel lines
the slopes of perpendicular lines
the midpoint formula
the distance formula

Coordinate Plane

The coordinate plane or Cartesian plane is a basic concept for coordinate geometry. It describes a two-dimensional plane in terms of two perpendicular axes: x and y. The x-axis indicates the horizontal direction while the y-axis indicates the vertical direction of the plane. In the coordinate plane, points are indicated by their positions along the x and y-axes.

For example: In the coordinate plane below, point L is represented by the coordinates (–3, 1.5) because it is positioned on –3 along the x-axis and on 1.5 along the y-axis. Similarly, you can figure out the positions for the points M = (2, 1.5) and N = (–2, –3).

The following video shows how to plot points in the coordinate plane and how to determine the coordinates of points on the coordinate plane.

On the coordinate plane, the slant of a line is called the slope. Slope is the ratio of the change in the y-value over the change in the x-value, also called rise over run.

Given any two points on a line, you can calculate the slope of the line by using this formula:

slope =

For example: Given two points, P = (0, –1) and Q = (4,1), on the line we can calculate the slope of the line.

slope = =

coord plane: slope

How to Find the Slope Given 2 Points.

Y-intercept

The y-intercept is where the line intercepts (meets) the y-axis.

For example: In the above diagram, the line intercepts the y-axis at (0,–1). Its y-intercept is equals to –1.

Equation Of A Line

In coordinate geometry, the equation of a line can be written in the form, y = mx + b, where m is the slope and b is the y-intercept. (see a mnemonic for this formula)

equation of a line

For example: The equation of the line in the above diagram is:

How to Write a Slope Intercept Equation for a Line on a Graph.

Negative Slope

Let's look at a line that has a negative slope.

For example: Consider the two points, R(–2, 3) and S(0, –1) on the line. What would be the slope of the line?

slope = =

negative slope

The y-intercept of the line is –1. The slope is –2. The equation of the line is:
y = –2x – 1

This video explains how to determine the slope of a line given the graph of a line with a negative slope.

Slopes Of Parallel Lines

In coordnate geometry, two lines are parallel if their slopes (m) are equal.

parallel lines

For example: The line is parallel to the line . Their slopes are both the same.

Write the equation of a line that is parallel to the line 2x - 4y = 8 and goes through the point (3, 0)

Slopes Of Perpendicular Lines

In the coordinate plane, two lines are perpendicular if the product of their slopes (m) is –1.

perpendicular

For example: The line is perpendicular to the line y = –2x – 1. The product of the two slopes is

Find the slope of the line that is perpendicular to the line 3x + 2y = 6.

Midpoint Formula

Some coordinate geometry questions may require you to find the midpoint of line segments in the coordinate plane. To find a point that is halfway between two given points, get the average of the x-values and the average of the y-values.

The midpoint between the two points (x₁,y₁) and (x₂,y₂) is

For example: The midpoint of the points A(1,4) and B(5,6) is

This video gives the formula for finding the midpoint of two points and one example to find the midpoint.

Distance Formula

In the coordinate plane, you can use the Pythagorean Theorem to find the distance between any two points.

The distance between the two points (x₁,y₁) and (x₂,y₂) is

distance

For example: To find the distance between A(1,1) and B(3,4), we form a right angled triangle with as the hypotenuse. The length of = 3 – 1 = 2. The length of = 4 – 1 = 3. Applying Pythagorean Theorem: