13.18 Perpendicular lines (Extended)
Perpendicular lines
Lines that meet at right angles ($90^\circ$90°) are called perpendicular lines.
Play with this applet creating pairs of perpendicular lines.
Fill in this table as you go.
| Gradient of line 1 | $m_1$m1 | |||
|---|---|---|---|---|
| Gradient of line 2 | $m_2$m2 | |||
| Product of line 1 and line 2 | $m_1\times m_2$m1×m2 |
What do you notice about the product of the gradients of lines $1$1 and $2$2? (The pair of perpendicular lines)
You will have discovered the perpendicular lines have gradients whose product is equal to $-1$−1.
We say that $m_1$m1 is the negative reciprocal of $m_2$m2.
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$2 and $-\frac{1}{2}$−12
$\frac{3}{4}$34 and $-\frac{4}{3}$−43
$-10$−10 and $\frac{1}{10}$110
- 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 $-1$−1 then the lines are perpendicular.
Practice questions
Question 1
A line which passes through the point $\left(0,6\right)$(0,6) is perpendicular to $y=-3x+5$y=−3x+5.
Find the gradient of this perpendicular line.
State the equation of the perpendicular line.