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Parallel & Perpendicular Lines (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Jamie Wood

Updated on

DP Analysis & Approaches (AA): SLDP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

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Parallel Lines

How are the equations of parallel lines connected?

  • Parallel lines are always equidistant meaning they never intersect

  • Parallel lines have the same gradient

    • If gradient of line l subscript 1 is m subscript 1 and gradient of line l subscript 2 is m subscript 2 ,

      • m subscript 1 equals m subscript 2 space rightwards double arrow space space l subscript 1 space & blank l subscript 2 blank are blank parallel

      • l subscript 1 blank & blank l subscript 2 blank are blank parallel space rightwards double arrow space 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

Graph with x and y axes showing two parallel lines, labelled with equations y=m₁x+c and y=m₂x+c, indicating m₁=m₂ to be parallel.

Worked Example

The line space l  passes through the point space left parenthesis 4 comma negative 1 right parenthesis  and is parallel to the line with equation space 2 x minus 5 y equals 3 .

Find the equation of space l , giving your answer in the form space y equals m x plus c.

2-1-1-ib-ai-sl-parallel-lines-we-solution

 

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Perpendicular Lines

How are the equations of perpendicular lines connected?

  • Perpendicular lines intersect at right angles

  • The gradients of two perpendicular lines are negative reciprocals

    • If gradient of line l subscript 1 is m subscript 1 and gradient of line l subscript 2 is m subscript 2 ,

      • 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

    • For example, if m subscript 1 equals 1 third and m subscript 2 equals negative 3 then the lines are perpendicular

      • If m subscript 1 equals 5 and m subscript 2 equals 1 fifth then the lines are not perpendicular

  • To determine if two lines are perpendicular:

    • Rearrange into the gradient-intercept form, space y equals m x plus c

    • Compare the values of m for each

    • If the product of the two values of m 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

Graph showing perpendicular lines intersecting at a right angle, labelled with equations y=m_1x+c and y=m_2x+c. m_1 times m_2 = -1.

Worked Example

The line space l subscript 1  is given by the equation space 3 x minus 5 y equals 7.

The line space l subscript 2  is given by the equation space y equals 1 fourth minus 5 over 3 x .

Determine whether space l subscript 1 and space l subscript 2 are perpendicular. Give a reason for your answer.

2-1-1-ib-ai-sl-perpendicular-lines-we-solution

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Jamie Wood

    Reviewer: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.