Perpendicular Lines (AQA GCSE Maths: Higher): Revision Note

Perpendicular lines

What are perpendicular lines?

• Perpendicular lines are straight lines which meet at right-angles (90°)

• One line may be referred to as a normal to the other line

How are the gradients of perpendicular lines related?

• Gradients m₁ and m₂ are perpendicular if m₁ × m₂ = −1

  • For example

    • 1 and −1

    • $\frac{1}{3}$ and -3

    • $-\frac{2}{3}$ and $\frac{3}{2}$

• The two gradients are negative reciprocals of one another

• We can use $m_2=-\frac{1}{m_1}$to find a perpendicular gradient

How can I tell if two lines are perpendicular?

• Given two lines in the form $y=mx+c$, simply check if their gradients $\left(m\right)$ are negative reciprocals of one another

  • $y=\frac{1}{3}x+10$ and $y=-3x-18$ are perpendicular

  • $y=\frac{1}{7}x+16$ and $y=7x-8$ are not perpendicular

• One or both of the equations may not be written in the form $y=mx+c$

  • In this case, you should rearrange both equations into the form $\mathit{y}\mathbf{=}\mathit{m}\mathit{x}\mathbf{+}\mathit{c}$

  • Their gradients can then be easily compared

How do I find the equation of a line perpendicular to another?

• You need to be able to find the equation of line that passes through a particular point and is perpendicular to another line

  • E.g. $5y=4x+30$ which passes through the point (8, 3)

• Rearrange the equation into the form $y=mx+c$ so that its gradient can be identified more easily

  • $y=\frac{4}{5}x+6$

• Find the gradient of the perpendicular line

  • The gradient of the original line is $\frac{4}{5}$

  • Therefore the gradient of the perpendicular line is $-\frac{5}{4}$

  • The perpendicular line has an equation in the form $y=-\frac{5}{4}x+c$

• Substitute the given point into the equation for the perpendicular and solve for $c$

  • Substitute (8, 3), into $y=-\frac{5}{4}x+c$

  • $3=-\frac{5}{4}\left(8\right)+c$

  • $c=13$

• Substitute the value of $c$ to find the equation of the perpendicular

  • The equation of the perpendicular line is $y=-\frac{5}{4}x+13$

    • This could also be written as $4y=-5x+52$ or equivalent

How do I find the equation of a perpendicular bisector?

• A perpendicular bisector of a line segment cuts the line segment in half at a right angle

• Finding the equation of the perpendicular bisector of a line segment is very similar to finding the equation of a any perpendicular

  • Find the coordinates of the midpoint of the line segment

    • The perpendicular bisector will pass through this point

  • Find the gradient of the line segment

  • Then find the negative reciprocal of this gradient

    • This will be the gradient of the perpendicular bisector, $m$

  • Write the equation of the perpendicular bisector in the form $y=mx+c$

  • Substitute the midpoint of the line segment into the equation of the perpendicular bisector

    • Solve to find $c$

  • Write the full equation of the perpendicular bisector in the form $y=mx+c$

  • Rearrange the equation if the question requires a different form