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Parallel & Perpendicular Lines

Parallel Lines

What are parallel lines?

  • Parallel lines are lines that have the same gradient, but are not the same line
  • Parallel lines do not intersect with each other
  • You can easily spot that two lines are parallel when they are written in the form y equals m x plus c, as they will have the same value of m (gradient)
    • y equals 3 x plus 7 and y equals 3 x minus 4 are parallel (and therefore will never intersect)
    • y equals 2 x plus 3 and y equals 3 x plus 3 are not parallel (and therefore must intersect)
    • y equals 4 x plus 9 and y equals 4 x plus 9 are not parallel; they are the exact same line

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

  • As parallel lines have the same gradient, a line of the form y equals m x plus c will be parallel to a line in the form y equals m x plus d, where m is the same for both lines
    • If c equals d then they would be the same line and therefore not parallel
  • If you are asked to find the equation of a line parallel to y equals m x plus c, you will also be given some information about a point that the parallel line, y equals m x plus d passes through; open parentheses x subscript 1 space comma space y subscript 1 close parentheses
  • You can then substitute this point into y equals m x plus d and solve to find d

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?

  • You should already know that parallel lines have equal gradients
  • Perpendicular lines meet each other at right angles – ie they meet at 90°

What’s the deal with perpendicular gradients (and lines)?

  • Before you start trying to work with perpendicular gradients and lines, make sure you understand how to find the equation of a straight line – that will help you do the sorts of questions you will meet
  • Gradients m1 and m2 are perpendicular if m1 × m2 = −1
    • For example
      • 1 and −1
      • 1 third and 3
      • negative 2 over 3 and 3 over 2
  • We can use m2 = −1 ÷ m1 to find a perpendicular gradient. This is called the negative reciprocal.
  • If in doubt, SKETCH IT!

Worked example

The line L has equation y equals 2 x minus 2
Find an equation of the line perpendicular to L which passes through the point open parentheses 2 comma space minus 3 close parentheses.
Leave your answer in the form a x plus b y plus c equals 0 where ab and 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 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 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 ab and c are integers

table 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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