3.01 Parallel and perpendicular lines
The equations y = 2 x, y = 2 x + 5 and \\y = 2 x - 7 have been graphed on the same number plane:
What do all of the equations have in common?
What do all the lines have in common?
State whether the following pairs of lines are parallel:
y = - 3 x - 2 and y = - 3 x + 9
y = - 2 x - 5 and y = - 2 x - 8
y = 7 x + 8 and y = - 5 x + 8
y = 4 x-1 and y = 4 x - 6
x=4 and y = 5
y = 7 x - 5 and y = - 7 x + 6
y = 3 x - 5 and y = -\dfrac{1}{3} x + 6
y=-8x-2 and 9x+7
State whether the following lines are parallel to y = 7 x + 3:
y = 7 x - 3
y = 6 x + 3
y = 7 x + 4
y = 7 x
y = 6 x + 4
y = - 7 x + 3
y = 3x + 7
y = - 3 x - 7
State whether the following lines are parallel to y = - 3 x + 2:
y = 3 x
- 3 y - x = 5
y = - 10 - 3 x
y + 3 x = 7
State whether the following lines are parallel to y = 9 x + 2:
y = 9 x
y = -9 x + 5
y = - 9 x + 2
y = 9 x - 2
Consider the lines L_{1}: 5 x + 8 y - 10 = 0 and L_{2}: y = - \dfrac{5 x}{8} + 8.
Find the gradient of line L_{1}.
Find the gradient of line L_{2}.
Are the two lines parallel?
Consider the line y = 2 x + 2. If every point on the line is shifted 2 units up, find the equation of the new line.
Every point on a particular line is shifted 6 units down. The resulting line has equation \\ y = 2 x - 2. Find the equation of the original line.
Find the equation of a line described by the following information:
Parallel to the x-axis and passes through \left( - 10 , 2\right).
Parallel to the y-axis and passes through \left( - 7 , 2\right).
Parallel to the line y = - 3 x - 8 and cuts the y-axis at - 4.
Parallel to the line y = 8 x - 3 and cuts the y-axis at 5.
Parallel to the line y = - 2 x + 9 and passes through the point \left( - 3 , 1\right).
If the following pairs of lines are parallel, find the value of b:
y=b x + 4 \\ 2y=\dfrac{4}{5} x - 7
The line L_{1} goes through \left(3, 2\right) and \left( - 2 , 4\right).
Find the gradient of line L_{1}.
Find the equation of the line that has a y-intercept of 1 and is parallel to line L_{1}.
The line L_{1} goes through \left( - 9 , 10\right) and \left(2, - 1 \right).
Find the gradient of line L_{1}.
Find the equation of the line that passes through \left(10, 8\right) and is parallel to line L_{1}.
The line L_{1} passes through the point \left(9, - 5 \right) and is parallel to the line y = - 5 x + 2.
Find the gradient of line L_{1}.
Find the equation of line L_{1}.
Describe what it means for two lines to be perpendicular.
If two lines are perpendicular, state the product of their gradients.
State whether the following pairs of lines are perpendicular:
y = x + 1 and y = x - 1
y = 3 x - 7 and y = - 3 x + 6
y = - 3 x + 6 and y = \dfrac{x}{3} + 7
y = \dfrac{2 x}{3} + 4 and y = -1 - \dfrac{3 x}{2}
y = - \dfrac{x}{3} - 9 and y = - \left( 3 x - 27 \right)
y = 3 x - 5 and y = -\dfrac{1}{3} x + 6
y=-\dfrac{2x}{3}+1 and y=-1+\dfrac{3x}{2}
y=\dfrac{3x}{5}+6 and y=\dfrac{-5x-30}{-3}
Given that the following pairs of lines are perpendicular, calculate the value of m:
5 x + 4 y + 8=0 \\m x + 9 y - 8=0
- 6 x + 6 y - 8=0 \\ m x + 10 y + 8=0
Find the gradient of the line perpendicular to the following lines:
Line with gradient 6
Consider the lines L_{1}: 8 x = - 7 y + 3 and L_2: 7 y - 8 x - 7 = 0.
Find the gradient of line L_{1}.
Find the gradient of line L_{2}.
Are the two lines perpendicular?
Find the equation of a line described by the following information:
Perpendicular to the x-axis and passes through \left( - 8 , - 1 \right).
Perpendicular to the y-axis and passes through \left( - 8 , - 8 \right).
Perpendicular to y = - \dfrac{x}{2} + 5, and goes through the point \left(0, 6\right).
Perpendicular to y = 6 x + 10, and has the same y-intercept.
Perpendicular to the line that passes through \left(1, 1\right) and \left(3, 13\right), and has a y-intercept of \left(0, - 4 \right).
The line L_{1} is perpendicular to y = 5 x - 4 and cuts the y-axis at 1.
Find the gradient of line L_{1}.
Find the equation of line L_{1}.
The line L_{1} is perpendicular to y = 10 x + 8 and passes through the point of intersection of the lines y = x + 5 and 8 x - 10 y + 60 = 0.
Find the gradient of line L_{1}.
Find the point of intersection of y = x + 5 and 8 x - 10 y + 60 = 0.
Find the equation of the perpendicular line L_{1}.
Consider the following points on the number plane:
Point A \left(2, - 1 \right)
Point B \left(4, - 7 \right)
Point C \left( - 3 , 1\right)
Point D \left( - 6 , 10\right)
Find the gradient of the line AB.
Find the gradient of the line CD.
Is the line CD parallel to AB?
Consider the following points on the number plane:
Point P \left(0, - 1 \right)
Point Q \left(5, 0\right)
Point R \left(0, 6\right)
Point S \left( - 5 , 5\right)
Find the gradient of PQ.
Find the gradient of RS.
Are PQ and RS parallel?
Are QR and PS parallel?
Consider the rhombus ABCD on the number plane:
Find the gradients of its diagonals.
Gradient of AC
Gradient of BD
Are the diagonals of the rhombus perpendicular? Explain your answer.
The four vertices of quadrilateral ABCD have been plotted on the number plane:
Find the gradient of the following sides:
AB
BC
DC
AD
Given that side AB = BC, determine the type of quadrilateral described by the four points.
Determine whether the following sets of points are collinear:
Point A \left( - 4 , 3\right), Point B \left( - 2 , 7\right) and Point C \left( - 7 , - 3 \right).
Point A \left( - 3 , 1\right), Point B \left( - 2 , 6\right) and Point C \left(0, 14\right).
Point A \left( - 2 , 4\right), Point B \left( 2 , 2 \right) and Point C \left( 4 , 1\right).