Parallel and perpendicular lines
On a map of Dublin, the Luas tracks and a nearby street can run in the same direction (they are parallel) or cross at a right angle (they are perpendicular). To find out without having to draw a map, all you need to do is look at one number for each line: its slope. We will answer this at the end of the page.
The slope of a line
Calculate the gradient of the line passing through points $A(2, -1)$ and $B(5, 8)$, and deduce from it what the gradient of any line perpendicular to it would be.
Calculate the gradient of the line passing through points $A(2, -1)$ and $B(5, 8)$, and deduce from it what the gradient of any line perpendicular to it would be.
Parallel lines
Given the lines $r: y = -2x + 5$ and $s: y = \frac{1}{2}x - 3$, state with reasons whether they are parallel, perpendicular, or intersecting but non-perpendicular.
Given the lines $r: y = -2x + 5$ and $s: y = \frac{1}{2}x - 3$, state with reasons whether they are parallel, perpendicular, or intersecting but non-perpendicular.
Find the explicit equation of the line that is parallel to $y = -3x + 2$ and passes through the point $P(0, 5)$.
Find the explicit equation of the line that is parallel to $y = -3x + 2$ and passes through the point $P(0, 5)$.
Write down the general equation of the line parallel to the x-axis passing through point $Q(-3, 8)$.
Write down the general equation of the line parallel to the x-axis passing through point $Q(-3, 8)$.
Perpendicular lines
Write down the gradient of a line that is parallel to and the gradient of a line that is perpendicular to the line with equation $y = 4x - 7$.
Write down the gradient of a line that is parallel to and the gradient of a line that is perpendicular to the line with equation $y = 4x - 7$.
Determine the point-slope equation of the line perpendicular to $y = \frac{2}{5}x - 1$ passing through the point $A(4, 2)$.
Determine the point-slope equation of the line perpendicular to $y = \frac{2}{5}x - 1$ passing through the point $A(4, 2)$.
Prove analytically that the quadrilateral whose vertices are $A(1, 1)$, $B(5, 3)$, $C(4, 5)$ and $D(0, 3)$ is a rectangle. To do this, check the gradients of its sides…
Prove analytically that the quadrilateral whose vertices are $A(1, 1)$, $B(5, 3)$, $C(4, 5)$ and $D(0, 3)$ is a rectangle. To do this, check the gradients of its sides and confirm that adjacent angles are right angles.
Parallelism and perpendicularity in general form
Check analytically whether the lines given in general form $r: 3x - 6y + 2 = 0$ and $s: x - 2y - 5 = 0$ are parallel.
Check analytically whether the lines given in general form $r: 3x - 6y + 2 = 0$ and $s: x - 2y - 5 = 0$ are parallel.
Calculate the value of parameter $k$ so that the line $kx - 3y + 6 = 0$ is parallel to the line joining points $A(1, 2)$ and $B(4, 11)$.
Calculate the value of parameter $k$ so that the line $kx - 3y + 6 = 0$ is parallel to the line joining points $A(1, 2)$ and $B(4, 11)$.
Determine the value of parameter $m$ so that lines $r: mx + 4y - 1 = 0$ and $s: 2x - 5y + 8 = 0$ are perpendicular.
Determine the value of parameter $m$ so that lines $r: mx + 4y - 1 = 0$ and $s: 2x - 5y + 8 = 0$ are perpendicular.
Direction vectors and other forms
A direction vector $(p, q)$ of a line gives slope $m=\dfrac{q}{p}$. Parallel lines have proportional direction vectors; a perpendicular direction is $(-q, p)$. A line through $(x_0,y_0)$ with direction $(p,q)$ has parametric form $x=x_0+pt,\ y=y_0+qt$ and symmetric form $\dfrac{x-x_0}{p}=\dfrac{y-y_0}{q}$.
A direction vector of line $r$ is $\vec{v} = (3, -1)$. Write down a direction vector for a line parallel to $r$ and another for a line perpendicular to $r$.
A direction vector of line $r$ is $\vec{v} = (3, -1)$. Write down a direction vector for a line parallel to $r$ and another for a line perpendicular to $r$.
Two aeroplanes fly at the same altitude. The route of the first aeroplane is given by the parametric equations $x = 1 + 5t$, $y = -2 - 3t$. The route of the second…
Two aeroplanes fly at the same altitude. The route of the first aeroplane is given by the parametric equations $x = 1 + 5t$, $y = -2 - 3t$. The route of the second aeroplane is parallel to that of the first and passes through the control tower $T(0, 0)$. Write down the parametric equations of the second aeroplane's route.
A main irrigation pipe follows the line given by the symmetric equation $\frac{x-2}{4} = \frac{y+1}{-3}$. A secondary pipe perpendicular to the main line needs to be…
A main irrigation pipe follows the line given by the symmetric equation $\frac{x-2}{4} = \frac{y+1}{-3}$. A secondary pipe perpendicular to the main line needs to be installed to connect to a sprinkler located at $S(-1, 5)$. Find the symmetric equation of this secondary pipe.
Applications: the perpendicular bisector
The perpendicular bisector of a segment $AB$ is the line perpendicular to $AB$ that passes through its midpoint. To find it, you combine two ideas you already know: the midpoint and the perpendicular slope.
The same idea gives the orthocentre of a triangle: find the equations of two altitudes (each through a vertex, perpendicular to the opposite side) and solve them as simultaneous equations.
Back to the start: two lines on a map are parallel if their slopes are equal and perpendicular if the product of their slopes is $-1$.
Find the explicit equation of the perpendicular bisector of the segment with endpoints $A(-2, 3)$ and $B(4, 1)$. (Remember that the perpendicular bisector is the line…
Find the explicit equation of the perpendicular bisector of the segment with endpoints $A(-2, 3)$ and $B(4, 1)$. (Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint).
A ship sails in a straight line following the path $r: 3x + 4y - 24 = 0$. A lighthouse is located at point $F(7, 7)$. Calculate the shortest distance from the lighthouse…
A ship sails in a straight line following the path $r: 3x + 4y - 24 = 0$. A lighthouse is located at point $F(7, 7)$. Calculate the shortest distance from the lighthouse to the ship's path. (Hint: find the line perpendicular to $r$ that passes through $F$, find the point of intersection and calculate the distance between both points).
Find the exact coordinates of the orthogonal projection of the point $P(3, -2)$ onto the line $r: x - 2y + 5 = 0$. (Note: the orthogonal projection is the point of…
Find the exact coordinates of the orthogonal projection of the point $P(3, -2)$ onto the line $r: x - 2y + 5 = 0$. (Note: the orthogonal projection is the point of intersection of the line $r$ with the line perpendicular to it passing through $P$).