1. Basic Definitions
Distance Formula
Recall Pythagoras' Theorem:
Also on this page:
Gradient (slope)
Inclination
Parallel Lines
Perpendicular Lines
Need Graph Paper?
For a right-angled triangle with hypotenuse length c,
We use this to find the distance between any two points (x1, y1) and (x2, y2) on the cartesian plane:
The Cartesian Plane
The cartesian plane was named after Rene Descartes. It is also called the x-y plane.
See more about Descartes in Functions and Graphs.
The point (x2, y1) is at the right angle. We can see that:
- The distance between the points (x1, y1) and (x2, y1) is simply x2 − x1 and
- The distance between the points (x2, y2) and (x2, y1) is simply y2 − y1.
Using Pythagoras' Theorem we have the distance between (x1, y1) and (x2, y2) given by:
Example 1:
Find the distance between the points (3, -4) and (5, 7).
First, let's see this in LiveMath.
Now for the normal answer:
Example 2:
Find the distance between the points (3, -1) and (-2, 5).
Gradient (or slope)
The gradient of a line is defined as
In this triangle, the gradient of the line is given by:
In general, for the line joining the points (x1, y1) and (x2, y2):
We see from the diagram above, that the gradient (usually written m) is given by:
Example:
Find the slope of the line joining the points (-4, -1) and (2, -5).
Positive and Negative Slopes
In general, a positive slope indicates the value of the dependent variable increases as we go left to right:
[The dependent variable in the above graph is the y-value.]
A negative slope means that the value of the dependent variable is decreasing as we go left to right:
Here is an animation of this using LiveMath.
Inclination
We have a line with slope m and the angle that the line makes with the x-axis is α.
From trigonometry, we recall that the tan of angle α is given by:
Now, since slope is also defined as opposite/adjacent, we have:
This gives us the result:
tan α = m
Then we can find angle α using
α = arctan m
(That is, α = tan-1m)
This angle α is called the inclination of the line.
Example 1:
Find the inclination of the line with slope 2.
NOTE: The size of angle α is (by definition) only between 0° and 180°.
Example 2:
Find the slope of the line with inclination α = 137°.
Let's see Gradient and Inclination using LiveMath.
Parallel Lines
Lines which have the same slope are parallel.
If a line has slope m1 and another line has slope m2 then the lines are parallel if
m1 = m2
Here is a LiveMath animation showing that if the gradient stays the same and we only change the y-intercept, the lines are parallel.
Perpendicular Lines
If a line has slope m1 and another line has slope m2 then the lines are perpendicular if
m1 × m2= -1
In the example at right, the slopes of the lines are 2 and -0.5 and we have:
2 × -0.5 = -1
So the lines are perpendicular.
Let's see a LiveMath example:
Example:
A line l has slope m = 4.
a) What is the slope of a line parallel to l?
b) What is the slope of a line perpendicular to l?
Special Cases
What if one of the lines is parallel to the y-axis?
For example, the line y = 3 is parallel to the x-axis and has slope 0. The line x = 3.6 is parallel to the y-axis and has an undefined slope.
The lines are clearly perpendicular, but we cannot find the product of their slopes. In such a case, we cannot draw a conclusion from the product of the slopes, but we can see immediately from the graph that the lines are perpendicular.
The same situation occurs with the x- and y-axes. They are perpendicular, but we cannot calculate the product of the 2 slopes, since the slope of the y-axis is undefined.
Exercises
- What is the distance between (-1, 3) and (-8, -4)?
- A line passes through (-3, 9) and (4, 4). Another line passes through (9, -1) and (4, -8). Are the lines parallel or perpendicular?
- Find k if the distance between (k,0) and (0, 2k) is 10 units.
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