Perpendicular Lines (Cambridge International IGCSE Mathematics)

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 $( m )$ 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$y=mx+c$
  • 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 $y=-\frac{5}{4}x+c$
  • $3 = - \frac{5}{4} ( 8 ) + c$
  • $c=13$

• 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

Worked Example The line L has equation $y-2x+2=0$.

Find the equation of the line perpendicular to L which passes through the point $( 2 , - 3 )$.

Leave your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

Answer:

*Rearrange L into the form *$y=mx+c$ so we can identify the gradient

$\begin{gathered} y - 2 x + 2 & = 0 \\ y = 2 x - 2 \end{gathered}$

Gradient of L is 2

The gradient of the line perpendicular to L will be the negative reciprocal of 2

$m=-\frac{1}{2}$

Substitute the point $( 2 , - 3 )$ into the equation $y=-\frac{1}{2}x+c$* * Solve for $c$

$\begin{gathered} y = - 1 \frac{1}{2} x + c \\ - 3 & = - 1 \frac{1}{2} ( 2 ) + c \\ - 3 & = - 1 + c \\ c = - 2 \end{gathered}$

Write the full equation of the line

$\begin{gathered} y = - 1 \frac{1}{2} x - 2 \end{gathered}$

The question asks for the line to be written in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers

Move all the terms to the left hand side

$\frac{1}{2}x+y+2=0$

Then multiply every term by 2, to ensure they are all integers

$x+2y+4=0$

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

Worked Example Find the equation of the perpendicular bisector of the line segment joining the points (4, -6) and (8, 6).

Answer:

Find the coordinates of the midpoint of the line segment The perpendicular bisector will pass through this point

$( \frac{4 + 8}{2} , \frac{- 6 + 6 }{2} ) = ( 6 , 0 )$

Find the gradient of the line segment

$\frac{-6-6}{4-8}=\frac{-12}{-4}=3$

Find the negative reciprocal of this This will be the gradient of the perpendicular bisector, $m$

$m=-\frac{1}{3}$

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

$y=-\frac{1}{3}x+c$

Substitute in the midpoint (6, 0) and solve to find $c$

$\begin{gathered} 0 = - 1 \frac{1}{3} ( 6 ) + c \\ 0 = - 2 + c \\ c = 2 \end{gathered}$

Write the full equation of the perpendicular bisector

$y=-\frac{1}{3}x+2$