Coordinate Geometry
The coordinate 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
why the points M = (2, 1.5) and N = (–3, –2).

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.
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 =
=

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)

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


The y-intercept of the line is –1. The slope is –2. The equation of the line
is:
y = –2x – 1
In coordnate geometry, two lines are parallel if their slopes (m) are equal.

For example: The line
is parallel to the line
. Their slopes are both the same.
In the coordinate plane, two lines are perpendicular if the product of their slopes (m) is –1.

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


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:
2 = 22 + 32
2 = 13
=
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