Slopes of Perpendicular Lines Questions

A detailed tutorial on how to prove that the slopes of two perpendicular lines are the negative reciprocal of each other. Questions on perpendicular lines are also included with their detailed solutions and explanations.

Prove that two perpendicular lines have slopes that are negative reciprocal of each other

  • The figure below shows two perpendicular lines L1 and L2.

    parallel lines and equal slopes. .

  • Let the equation of line L1 be
    y = m1 x + b1
  • We now select two points A and B on line L1 whose x coordinates are 6 and 8 respectively and find their y coordinates.
    A(6 , 6 m1 + b1)
    B(7 , 7 m1 + b1)
  • We now find the components of vector AB.
    vector(AB) = > 7 - 6 , (7 m1 + b1) - (6 m1 + b1) >
    = > 1 , m1>
  • Let y = m2 x + b2 be the equation of line L2. We now select two points C and D on line L2 whose x coordinates are 4 and 5 respectively and find their y coordinates.
    A(4 , 4 m2 + b2)
    B(5 , 5 m2 + b2)
    vector(CD) = > 5 - 4 , (5 m2 + b2) - (4 m2 + b2) >
    = > 1 , m2>
  • If the two lines are perpendicular then the two vectors AB and CD are orthogonal and their scalar product must be zero. Hence
    > 1 , m1> . > 1 , m2> = 0
  • Which gives
    1 + m1 m2 = 0
  • The above may be written as
    m1 m2 = - 1 , the product of the slopes of two perpendicular lines is equal to - 1.
    m1 = - 1 / m2 , m1 is equal to the negative reciprocal of m2
    m2 = - 1 / m1 , m2 is equal to the negative reciprocal of m1
  • NOTE: Points A, B, C and D and their x coordinates can chosen anywhere on the two lines. As an exercise choose other values for the x coordinates of points A, B, C and D and redo the calculation above. You should be able to make the same conclusion concerning the slopes of two perpendicular lines.

Questions on perpendicular lines

Question 1
Which of the lines given by the equations,
a) y = - 2 x - 3    b) - 4 y + 5 x = 4    c) (1 / 2) x - y = 2    d) - 6 y - 3 x = 9
are perpendicular?
Solution to Question 1
We first need to write each line in slope intercept form, y = m x + b and find its slope m.
a) y = - 2 x - 3, slope = - 2
b) - 4 y + 5 x = 4, solve for y, y = (5 / 4) x - 1, slope = 5 / 4
c) (1 / 2) x - y = 2, solve for y, y = (1 / 2) x - 2, slope = 1 / 2
d) - 6 y - 3 x = 9, solve for y, y = - (1 / 2) x - 3 / 2, slope = - 1 / 2
The product of the slopes of the line in part a) and c) is
(-2)(1 / 2) = - 1
No other pair of lines have slopes with product equal to - 1.
Therefore the lines in part a) and c) are perpendicular.


Question 2
Find k so that the line through the points (1 , - 2) and (0 , 3) and the line through the points (3 , - k ) and ( 2 , - 5) are perpendicular.
Solution to Question 2
Find the slope m1 through the points (1 , - 2) and (0 , 3)
m1 = (3 - (- 2)) /(0 - 1) = - 5
Find the slope m2 through the points (3 , - k ) and ( 2 , - 5)
m2 = (- 5 - (- k)) / (2 - 3) = (- 5 + k) / (- 1) = 5 - k
The two lines are perpendicular, therefore the product of their slopes m1 and m2 must be equal to - 1
- 5 (5 - k) = - 1
Solve for k
k = 24 / 5


Question 3
Find k so that the lines with equations - k x + 2 x - y = -3 and - 2 x + y = 7 are perpendicular.
Solution to Question 3
Solve for y each of the given equations and write it in slope intercept form y = m x + b then find its slope m.
- k x + 2 x - y = - 3 , solve for y, y = (- k + 2) x + 3 , slope = - k + 2
- 2 x + y = 7 , solve for y , y = 2 x + 7 , slope = 2
The two lines are perpendicular and therefore the product of their slopes is equal to - 1; hence
2 (- k + 2) = - 1
Solve the above equation for k to obtain
k = 5 / 2


Question 4
Find the coordinate of point C so that the points A(0 , 2) , B(2 , 6) and C(2 k , k) are the vertices of a right angled triangle with hypotenuse AC.
Solution to Question 4
For the points A, B and C to be the vertices of a right angled triangle with hypotenuse AC, segment AB must be perpendicular to segment BC.
slope of segment AB = (6 - 2) / (2 - 0) = 2
slope of segment BC = (k - 6) / (2 k - 2)
for the segments AB and BC to be perpendicular, the product of their slopes must be equal to - 1. Hence
2 (k - 6) / (2 k - 2) = - 1
cross multiply to obtain
2 (k - 6) = - 2 k + 2
We now solve the above equation for k to obtain.
k = 7 / 2
Coordinates of point C are given by
(2 k , k ) = (7 , 7 / 2)


Question 5
Find the equation of a line L1 that passes through the point (3 , 1) and is perpendicular to the line L2 with equation - x + 2 y - 6 = 0.
Solution to Question 5
In order to find the slope m2 of the line L2, we need to write its equation in slope intercept form y = m x + b. Hence solve - x + 2 y - 6 = 0 for y to obtain
y = (1 / 2) x + 3
Hence
m2 = 1 / 2
Let m1 be the slope of the line L1. L1 and L2 are perpendicular and therefore the product of their slopes is equal to -1. Hence
m1 (1 / 2) = - 1
Solve the above to obtain
m1 = - 2
We now write the equation of the line L1 in point ( 3 , 1) slope m1 form is given by
y - 1 = - 2(x - 3)
which simplifies to
y = - 2 x + 7


Question 6
Find the equation of a line L1 that passes through the point (- 2 , - 1) and is perpendicular to the line L2 through the points (0 , - 2) and ( 2 , 4).
Solution to Question 6
The slope m2 of the line L2 through the points (0 , - 2) and ( 2 , - 4) is given by
m2 = ( 4 - (-2)) / (2 - 0) = 3
Let m1 be the slope of the line L1. L1 and L2 being perpendicular the product of their slopes is equal to -1. Hence
m1 (3) = - 1
Solve the above to obtain m1 = - 1 / 3
The equation of line L1 in point ( - 2 , - 1) slope m1 form is given by
y - (-1) = ( - 1 / 3)(x - (-2))
which simplifies to
y = - (1 / 3) x - 5 / 3

More References and Links

Equations of Lines in Different Forms,
Slope of a Line
Tutorial on Equation of Line
Line Problems
Equations of Line Through Two Points And Parallel and Perpendicular.