Perpendicular lines - Intermediate and Higher tier
Perpendicular lines will always cross at right angles.
To determine if two lines are perpendicular, we need to multiply their gradients together. If the lines are perpendicular to each other, the product of their gradients will be -1.
We say \({M_{1} \times~M_{2} = -1}\), where \({M_{1}}\) is the gradient of the first line and \({M_{2}}\) is the gradient of the second line.
Example
Are the lines \(\text{y = x - 5}\) and \(\text{y + x = 3}\) perpendicular to each other?
First we need to ensure both lines are in the form \(\text{y = mx + c}\).
Rearranging \(\text{y + x = 3}\) gives us \(\text{y = -x + 3}\).
So, \({M_{1} = 1}\) and \({M_{2} = -1}\).
\({M_{1} \times ~M_{2} = 1 \times -1 = -1}\).
So the lines are perpendicular.
Question
Are the lines \(\text{2y = x + 7}\) and \(\text{y + 2x = 3}\) perpendicular?
Rearranging, we get \(\text{y = 0.5x + 3.5}\) and \(\text{y = -2x + 3}\).
So \({M_{1} = 0.5}\) and \({M_{2} = -2}\).
\({M_{1} \times~M_{2} = 0.5 \times -2 = -1}\).
So the lines are perpendicular.
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