Coordinate Geometry – The Basics

If we look at the below straight line graph we can pick two points anywhere along the line and use them to calculate a number of things.

Straight line graph

The first point has coordinates (x1 , y1) and the second point has coordinates (x2 , y2). These points can be substituted with any real coordinate on a line graph.

Midpoint

The midpoint of these two points can be calculated using the following equation:

Equation for calculating midpoint

Distance

The distance (d) between the two points can be found using this equation:

Equation for calculating distance

Gradient

The two points can also be used to calculate the gradient of the line. This can be found by finding the difference in the y coordinates and dividing them by the difference in the x coordinates, as shown by the following equation:

Equation for calculating gradient

Example

In the below graph two points are highlighted.The first point has the coordinates (3 , 3) and the second point has the coordinates (8 , 8).

Straight line graph

We can use these coordinates to calculate the midpoint between the two points, the distance between the two points and the gradient of the line. They can be substituted into the above equations where from the first point we know that x1 = 3 and y1 = 3 and from the second point we know that x2 = 8 and y2 = 8 (NOTE: you can swap the coordinates round so that x1 = 8 and y1 = 8 and x2 = 3 and y2 = 3  you just have to be consistent throughout your workings):

The Equation of a Straight Line Graph

The general equation of a straight line graph can be written as:

Equation of a straight line

Where m is the gradient of the graph (calculated as shown above) and c is the y intercept of the graph (the point where it crosses the y axis).

Parallel Lines

Let’s take a look at two straight line graphs:

If the gradients of both of these lines are equal: m1 = m2 the lines are parallel to each other.

Perpendicular Lines

If the gradients of two lines multiply together to give -1: m1 x m2 = -1 the lines are perpendicular to each other. The gradients are the negative reciprocals of each other.

Equation of a Line Passing Through 2 Points

If we know the coordinates of two points on a straight line graph we can use them to calculate the equation of the line using the following equation:

The reason two points are needed is because we need to calculate the gradient. Once the gradient is calculated it can be substituted into this equation along with the coordinates of one of the points. Once re-arranged you will get the equation of the line in the conventional form of y = mx + c.

However, if the gradient is known we only need one point on the line which can be substituted in.

Worked Examples

Example 1

Two points are marked on the graph below. Find the midpoint of these points and calculate the distance between these two points.

Graph for example 1

Solution to Example 1

Example 2

(7, 14) and (10 , 23) both lie on a straight line graph. Find the equation of the line and also find the equation of the line that is perpendicular and passes through the origin.

Solution to Example 2

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