Slopes of Parallel Lines Questions

Slopes of Parallel Lines Questions

A detailed tutorial on how to prove that the slopes of two parallel lines are equal. Questions on parallel lines are also included with their detailed solutions.

Prove that two parallel lines have equal slopes

  • The figure below shows two parallel lines L1 and L2. Points A and B are on the line L1 and points C and D are on the line L2.

    parallel lines and equal slopes. .

  • Let BM and DN be parallel to the y axis and AM and CN parallel to the x axis. Triangles ABM and CDN have all their corresponding sides parallel are therefore similar. Hence
    BM / DN = MA / NC
  • The above may be written
    BM / MA = DN / NC
  • According to the definition of slope, BM / MA is the slope of line L1 and DN / NC is the slope of L2 and are therefore equal.

Questions on parallel lines

Question 1
Which of the lines given by the equations
a) y = 2 x - 3    b) 2 x - y = 2    c) - 4 y + 2 x = 0    d) - 4 y + 8 x = 9
are parallel?
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) 2 x - y = 2, solve for y, y = 2 x - 2, slope = 2
c) - 4 y + 2 x = 0, solve for y, y = (1 / 2) x, slope = 1 / 2
d) - 4 y + 8 x = 9, solve for y, y = 2 x - 9 / 4, slope = 2
The lines with equal slopes are the lines given in parts a) b) and d) and they are therefore parallel.


Question 2
Find k so that the lines with equations 6 k x - 3 y = 9 and - 4 x + 5 y = 7 are parallel.
Solution to Question 2
Solve for y and find the slope of each line
6 k x - 3 y = 9 , solve for y, y = 2 k x - 3 , slope = 2 k
- 4 x + 5 y = 7 , solve for y , y = (4 / 5) x + 7 / 5 , slope = 4 / 5
The two lines are parallel and therefore their slopes are equal; hence
2 k = 4 / 5
Solve for k
k = 2 / 5


Question 3
Find a so that the line through the points (1 , 2) and (0 , 3) and the line through the points (a , 2) and ( -2 , 7) are parallel.
Solution to Question 3
Find the slope m1 through the points (1 , 2) and (0 , 3)
m1 = (3 - 2) /(0 - 1) = - 1
Find the slope m2 through the points (a , 2) and (-2 , 7)
m2 = (7 - 2) / (-2 - a) = 5 / (-2 - a)
The two lines are parallel, hence their slopes are equal
5 / (-2 - a) = - 1
Solve for a
a = 3


Question 4
Find the coordinate of point D so that the points A(0 , 2) , B(2,6), C(8 , 8) and D are the vertices of a parallelogram.
Solution to Question 4
Let (a , b) be the coordinates of point D. For the points A, B, C and D to be the vertices of a parallelogram, segment AB must be parallel to segment DC and segment BC must be parallel to segment AD as shown in the figure below.

parallelogram with vertices A, B, C and D.


slope of segment AB = (6 - 2) / (2 - 0) = 2
slope of segment DC = (8 - b) / (8 - a)
for the segments AB and DC to be parallel, their slopes must be equal, hence the equation
2 = (8 - b) / (8 - a)
cross multiply to obtain
8 - b = 16 - 2 a
slope of segment BC = (8 - 6) / (8 - 2) = 1 / 3
slope of segment AD = (b - 2) / (a - 0) = (b - 2) / a
For the segments BC and AD to be parallel, their slopes must be equal
1 / 3 = (b - 2) / a
cross multiply to obtain
a = 3 b - 6
We now solve the system of the two equations in a and b obtained above and given by
8 - b = 16 - 2 a
a = 3 b - 6
Substitute a by 3 b - 6 in the equation 8 - b = 16 - 2 a to obtain
8 - b = 16 - 2 (3 b - 6)
Solve the above equation for b to obtain
b = 4
then use any of the two equations to find a
a = 6
The coordinates of point D are
D(6 , 4)


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

Question 6
Find the equation of a line L1 that passes through the point (-1 , 1) and is parallel to the line L2 with equation 2 y + 4 x = 2.
Solution to Question 6
The slope m2 of the line L2 is found by first writing its equation in slope intercept form y = m x + b. Hence solve 2 y + 4 x = 2 for y to obtain
y = - 2 x + 1
The slope of L2 is
m2 = - 2
Since L1 and L2 are parallel their slopes are equal. Hence
m1 = m2 = - 2
The equation of line L1 in point (-1 , 1) slope m1 form is given by
y - 1 = - 2(x - (-1))
which simplifies to
y = - 2 x - 1

More References and Links

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

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