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By the end of this topic, you should be able to:
Perpendicular lines are two lines that meet (cross) at a 90° angle (a right angle). Think of the corner of a square or rectangle — those sides are perpendicular to each other.
When we work with straight lines on a graph, each line has a gradient (also called slope). The gradient tells us how steep the line is. For perpendicular lines, their gradients have a very special relationship.
If two lines are perpendicular to each other, their gradients are connected by this rule:
The gradient of the perpendicular line is the negative reciprocal of the original gradient.
Let's break this down:
So if the original gradient is m, the perpendicular gradient is –1/m.
If the gradient of line AB is m, then the gradient of a line perpendicular to AB is:
m⊥ = –1/m
(The symbol ⊥ means "perpendicular")
Example 1: If a line has gradient m = 2, what is the gradient of a perpendicular line?
Solution:
Answer: The perpendicular gradient is –1/2
Example 2: If a line has gradient m = –3/4, what is the gradient of a perpendicular line?
Solution:
Answer: The perpendicular gradient is 4/3
Example 3: Find the gradient of a line perpendicular to 2y = 3x + 1.
Solution:
First, we need to rearrange the equation into the form y = mx + c, where m is the gradient.
2y = 3x + 1
Divide everything by 2:
y = (3/2)x + 1/2
So the gradient of this line is m = 3/2.
Now find the perpendicular gradient:
Answer: The gradient of the perpendicular line is –2/3
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