Perpendicular Lines (Edexcel IGCSE Maths A (Modular))

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

    • and -3

    • and 

• The two gradients are negative reciprocals of one another

• We can use to find a perpendicular gradient

How can I tell if two lines are perpendicular?

• Given two lines in the form , simply check if their gradients are negative reciprocals of one another

  • and are perpendicular

  • and are not perpendicular

• One or both of the equations may not be written in the form

  • In this case, you should rearrange both equations into the form

  • 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. which passes through the point (8, 3)

• Rearrange the equation into the form so that its gradient can be identified more easily

  • 

• Find the gradient of the perpendicular line

  • The gradient of the original line is

  • Therefore the gradient of the perpendicular line is

  • The perpendicular line has an equation in the form

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

  • Substitute (8, 3), into

  • 

  • 

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

  • The equation of the perpendicular line is

    • This could also be written as 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,

  • Write the equation of the perpendicular bisector in the form

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

    • Solve to find

  • Write the full equation of the perpendicular bisector in the form

  • Rearrange the equation if the question requires a different form