Coordinate Geometry

Coordinate Plane

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).

Slopes

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 = =

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)

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

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 = =

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

Slopes Of Parallel Lines

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.

Slopes Of Perpendicular Lines

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

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
   

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
  

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 + 3^{2
    2} = 13
    =

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