1. Basic Definitions

Distance Formula

Recall Pythagoras' Theorem:

Also on this page:

Gradient (slope)
Inclination
Parallel Lines
Perpendicular Lines

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math expression

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.

LIVEMath

Now for the normal answer:

Answer

Example 2:

Find the distance between the points (3, -1) and (-2, 5).

Answer

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):

math expression

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).

Answer

Positive and Negative Slopes

In general, a positive slope indicates the value of the dependent variable increases as we go left to right:

math expression

[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:

math expression

Here is an animation of this using LiveMath.

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:

inclination

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.

Answer

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°.

Answer

Let's see Gradient and Inclination using LiveMath.

LIVEMath

Parallel Lines

math expression

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.

LIVEMath

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:

LIVEMath

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?

Answers

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.

Answers

Plane Analytical Geometry

2. The Straight Line

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