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