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Parallel & Perpendicular Lines (Cambridge O Level Additional Maths)

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Parallel Lines

What are parallel lines?

  • Parallel lines are always equidistant meaning they never intersect
  • Parallel lines have the same gradient
    • If the gradient of line l1 is m1 and gradient of line l2 is m2 then...
      • m subscript 1 equals m subscript 2 rightwards double arrow l subscript 1 blank & blank l subscript 2 blank are blank parallel
      • l subscript 1 blank & blank l subscript 2 blank are blank parallel rightwards double arrow m subscript 1 equals m subscript 2
  • To determine if two lines are parallel:
    • Rearrange into the gradient-intercept form space y equals m x plus c
    • Compare the coefficients of space x
    • If they are equal then the lines are parallel

Equations of parallel lines

Examiner Tip

  • Look for hidden parallel lines in an exam question
    • Parallel lines could be implied in an exam by phrases like “… at the same rate …”
    • Check the properties of a geometrical shape

Worked example

Find the equation of the line that is parallel to y equals 3 x plus 7 and passes through (2,1).

  

As the gradient is the same, the line that is parallel will be in the form: 

y equals 3 x plus d

Substitute in the coordinate that the line passes through: 

1 equals 3 open parentheses 2 close parentheses plus d

Simplify: 

1 equals 6 plus d

Subtract 6 from both sides: 

negative 5 equals d

Final answer: 

bold italic y bold equals bold 3 bold italic x bold minus bold 5

 

Perpendicular Lines

What are perpendicular lines?

  • Perpendicular lines intersect at right angles
  • The gradients of two perpendicular lines are negative reciprocals
    • If the gradient of line l1 is m1 and gradient of line l2 is m2 then...
      • space m subscript 1 cross times m subscript 2 equals negative 1 space rightwards double arrow space l subscript 1 space & space l subscript 2 space are space perpendicular
      •   l subscript 1 space & space l subscript 2 space are space perpendicular space rightwards double arrow space m subscript 1 cross times m subscript 2 equals negative 1
  • To determine if two lines are perpendicular:
    • Rearrange into the gradient-intercept form space y equals m x plus c
    • Compare the coefficients of space x
    • If their product is -1 then they are perpendicular
  • Be careful with horizontal and vertical lines
    • space x equals p and space y equals q are perpendicular where p and q are constants

Equations of perpendicular lines

Examiner Tip

  • Exam questions are good at “hiding” perpendicular lines
    • For example a tangent and a radius are perpendicular

Worked example

Find the equation of the line that is perpendicular to y equals 2 x minus 2 and passes through (2, -3).

Leave your answer in the form a x plus b y plus c equals 0 where a comma space b comma space c are integers.
 

L is in the form y equals m x plus c so we can see that its gradient is 2

m subscript 1 equals 2

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

m subscript 2 equals negative 1 half

Now we need to find c for the line we're after
Do this by substituting the point open parentheses 2 comma space minus 3 close parentheses into the equation y equals negative 1 half x plus c and solving for c

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 end cell equals cell negative 1 half cross times 2 plus c end cell row cell negative 3 end cell equals cell negative 1 plus c end cell row c equals cell negative 2 end cell end table

Now we know the line we want is 

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 1 half x minus 2 end cell end table

But this is not in the form asked for in the question. So rearrange into the form a x plus b y plus c equals 0 where a, b and c are integers

table attributes columnalign right center left columnspacing 0px end attributes row cell y plus 1 half x plus 2 end cell equals 0 row cell 2 y plus x plus 4 end cell equals 0 end table

Write the final answer

bold italic x bold plus bold 2 bold italic y bold plus bold 4 bold equals bold 0

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.