Parallel lines
Let's look at how we can identify parallel lines given their equations.
| Equation form | Characteristic of parallel lines | Examples |
|---|---|---|
| y=mx+c | \text{Parallel lines have the same }m\text { value.} | y=2x-1 \\ y=4+2x |
| ax+by+c=0 | \text{Parallel lines have the same value of } -\dfrac{a}{b} | x+2y-3=0 \\ 2x+4y+1=0 |
For every straight line y=mx+c, there exist infinitely many lines parallel to it.
Examples
Example 1
Which two lines are parallel to y=-3x+2?
Parallel lines have the same gradient.
Horizontal and vertical lines
Horizontal lines are lines where the y-value is always the same.
So if the equation of a straight line is y=c, then it will be a horizontal line passing through the point \left(0,c\right).
The x-axis itself is a horizontal line. The equation of the x-axis is y=0.
Vertical lines are lines where the x-value is always the same.
So if the equation of a straight line is x=c, then it will be a horizontal line passing through the point \left(c,0\right).
The y-axis itself is a horizontal line. The equation of the y-axis is x=0.
Examples
Example 2
Write down the equation of a line that is parallel to the x-axis and passes through \left(-10,2\right).
All horizontal lines are parallel to the x-axis and are of the form y=c.
Horizontal lines have a gradient of 0.
All vertical lines are parallel to the y-axis and are of the form x=c.
Vertical lines have an undefined gradient.
Perpendicular lines
Lines that meet at right angles (90\degree) are called perpendicular lines.
Exploration
Play with this applet to create pairs of perpendicular lines.
Fill in this table as you go.
| Gradient of line 1 | m_{1} | \\ | \\ | \\ |
|---|---|---|---|---|
| Gradient of line 2 | m_{2} | \\ | \\ | \\ |
| Product of line 1 and line 2 | m_{1} \times m_{2} | \\ | \\ | \\ |
What do you notice about the product of the gradients of lines 1 and 2? (The pair of perpendicular lines)
The product of the gradients of perpendicular lines will always be equal to -1.
Perpendicular lines have gradients whose product is equal to -1. So m_1 \times m_2 = -1.
We say that m_{1} is the negative reciprocal of m_{2}.
Negative reciprocal is a complex sounding term, but it just means two numbers that have opposite signs and are reciprocals of each other.
Here are some examples of negative reciprocals:
2 \ \text{ and } -\dfrac{1}{2} , \quad \quad \dfrac{3}{4} \ \text{ and } -\dfrac{4}{3}, \quad \quad \dfrac{3}{4} \ \text{ and } -\dfrac{4}{3}
Examples
Example 3
Find the equation of a line that is perpendicular to y=6x+10, and has the same y-intercept.
Give your answer in the form y=mx+c.
Two lines are perpendicular if their gradients are negative reciprocals of each other.
To test if lines are perpendicular multiply the gradients together. If the result is -1 then the lines are perpendicular.