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Parallel Lines and Perpendicular Lines

7.2 Parallel Lines and Perpendicular Lines
 
The image features a title that reads ‘Parallel Lines and Perpendicular Lines.’ Below the title, there are two boxes with descriptions: 1. The left box is labeled ‘Parallel Lines’ and contains the text: ‘Two lines are parallel if they have the same slope (gradient) and never intersect.’ 2. The right box is labeled ‘Perpendicular Lines’ and contains the text: ‘Two lines are perpendicular if the product of their slopes (gradients) is -1.’ There is also a logo for ‘Pandai’ positioned between the two boxes.
 
Parallel Lines
Figure

Two parallel lines with arrows pointing right, each with gradient m_1 and m_2.
Gradient, \(m\)
  • Parallel lines have equal gradients: \(m_1=m_2\).
  • Example:
    The lines \(y=2x+3\) and \(y=2x-4\) are parallel since they have the same gradient, \(m=2\).
Equation of Parallel Lines
  • Given a line with equation \(y=mx+c\), any line parallel to it can be written as \(y=mx+c_1\), where \(c_1\) is a different constant.
Properties
  • Parallel lines are always equidistant from each other.
  • No solution exists for a system of equations representing parallel lines (they do not intersect).
 
Perpendicular Lines
Figure

An illustration featuring two perpendicular lines that cross in the middle, visually representing the m_1 and m_2 gradients.
Gradient Relationship
  • If line \(1\) has a gradient \(m_1\) and line \(2\) has a gradient \(m_2\), then \(m_1 \times m_2=-1\).
  • Example:
    The lines \(y=2x+3\) and \(y=-\dfrac{1}{2}x +5\) are perpendicular because \(2\times\left(-\dfrac{1}{2}\right)=-1\).
Equation of Perpendicular Lines
  • Given a line with gradient, \(m\), a line perpendicular to it will have a gradient of \(-\dfrac{1}{m}\).
Properties
  • Perpendicular lines intersect at a \(90^\circ\) angle.
  • At the point of intersection, the gradients of the two lines multiply to \(-1\).
 
Finding Gradients
From two points

\(m=\dfrac{y_2-y_1}{x_2-x_1}\)
From equations
  • For a line in the form \(y=mx+c\), the gradient is \(m\).
  • For a line in the general form \(ax+by=c\), the gradient is \(m=-\dfrac{a}{b}\).
 
Examples
Parallel Lines
  • Given line \(1\): \(y=3x+2\), find the equation of a line parallel to it passing through \((1,4)\).
     
  • Solution:
    SInce the gradient must be the same, \(m=3\).
    Using \(y=mx+c\) with point \((1,4)\):
    \(\begin{aligned} 4&=3(1)+c \\ c&=1. \end{aligned}\)
    Equation of parallel line: \(y=3x+1\).
Perpendicular Lines
  • Given line \(1\): \(y=2x-5\), find the equation of a line perpendicular to it passing through \((2,3)\).
     
  • Solution:
    Gradient of perpendicular line \(m_2=-\dfrac{1}{2}\).
    Using \(y=mx+c\) with point \((2,3)\):
    \(\begin{aligned} 3&=-\dfrac{1}{2}(2)+c \\ c&=4. \end{aligned}\)
    Equation of peprpendicular line: \(y=-\dfrac{1}{2}x+4\).
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.

Parallel Lines and Perpendicular Lines

7.2 Parallel Lines and Perpendicular Lines
 
The image features a title that reads ‘Parallel Lines and Perpendicular Lines.’ Below the title, there are two boxes with descriptions: 1. The left box is labeled ‘Parallel Lines’ and contains the text: ‘Two lines are parallel if they have the same slope (gradient) and never intersect.’ 2. The right box is labeled ‘Perpendicular Lines’ and contains the text: ‘Two lines are perpendicular if the product of their slopes (gradients) is -1.’ There is also a logo for ‘Pandai’ positioned between the two boxes.
 
Parallel Lines
Figure

Two parallel lines with arrows pointing right, each with gradient m_1 and m_2.
Gradient, \(m\)
  • Parallel lines have equal gradients: \(m_1=m_2\).
  • Example:
    The lines \(y=2x+3\) and \(y=2x-4\) are parallel since they have the same gradient, \(m=2\).
Equation of Parallel Lines
  • Given a line with equation \(y=mx+c\), any line parallel to it can be written as \(y=mx+c_1\), where \(c_1\) is a different constant.
Properties
  • Parallel lines are always equidistant from each other.
  • No solution exists for a system of equations representing parallel lines (they do not intersect).
 
Perpendicular Lines
Figure

An illustration featuring two perpendicular lines that cross in the middle, visually representing the m_1 and m_2 gradients.
Gradient Relationship
  • If line \(1\) has a gradient \(m_1\) and line \(2\) has a gradient \(m_2\), then \(m_1 \times m_2=-1\).
  • Example:
    The lines \(y=2x+3\) and \(y=-\dfrac{1}{2}x +5\) are perpendicular because \(2\times\left(-\dfrac{1}{2}\right)=-1\).
Equation of Perpendicular Lines
  • Given a line with gradient, \(m\), a line perpendicular to it will have a gradient of \(-\dfrac{1}{m}\).
Properties
  • Perpendicular lines intersect at a \(90^\circ\) angle.
  • At the point of intersection, the gradients of the two lines multiply to \(-1\).
 
Finding Gradients
From two points

\(m=\dfrac{y_2-y_1}{x_2-x_1}\)
From equations
  • For a line in the form \(y=mx+c\), the gradient is \(m\).
  • For a line in the general form \(ax+by=c\), the gradient is \(m=-\dfrac{a}{b}\).
 
Examples
Parallel Lines
  • Given line \(1\): \(y=3x+2\), find the equation of a line parallel to it passing through \((1,4)\).
     
  • Solution:
    SInce the gradient must be the same, \(m=3\).
    Using \(y=mx+c\) with point \((1,4)\):
    \(\begin{aligned} 4&=3(1)+c \\ c&=1. \end{aligned}\)
    Equation of parallel line: \(y=3x+1\).
Perpendicular Lines
  • Given line \(1\): \(y=2x-5\), find the equation of a line perpendicular to it passing through \((2,3)\).
     
  • Solution:
    Gradient of perpendicular line \(m_2=-\dfrac{1}{2}\).
    Using \(y=mx+c\) with point \((2,3)\):
    \(\begin{aligned} 3&=-\dfrac{1}{2}(2)+c \\ c&=4. \end{aligned}\)
    Equation of peprpendicular line: \(y=-\dfrac{1}{2}x+4\).
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.
Chapter : Coordinate Geometry
Topic : Solve problems involving equations of parallel and perpendicular lines
Form 4 Additional Mathematics
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