Parallel and Perpendicular Lines
Look at a street map of a city grid. The avenues run side by side and never meet. The cross streets meet every avenue at a right angle. Then there is a diagonal boulevard that cuts across everything at some odd angle. Can you tell which streets are parallel, which are perpendicular, and which are neither, without drawing the map? You only need one number from each street: its slope.
The slope of a line
Slope is just a ratio, so it does not matter which two points of the line you pick. Some problems say gradient instead of slope. It is the same number.
Do the slope part of this exercise now. Its last part ("deduce the perpendicular slope") will make sense after the next two sections.
Calculate the gradient of the line passing through the 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 the 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
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 $P(0, 5)$.
Perpendicular lines
Drag the slider: the two lines stay at a right angle no matter how steep the first one is.
Write down the gradient of a line that is parallel 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 and the gradient of a line that is perpendicular to the line with equation $y = 4x - 7$.
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 not 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 not perpendicular.
Lines through a given point
Usually you know a slope and one point of the line, and you need its equation.
To find a line parallel to a given line, copy its slope. To find a line perpendicular to it, use the negative reciprocal. Then plug in the point.
Determine the point-gradient equation of the line perpendicular to $y = \frac{2}{5}x - 1$ that passes through the point $A(4, 2)$.
Determine the point-gradient equation of the line perpendicular to $y = \frac{2}{5}x - 1$ that passes through the point $A(4, 2)$.
A laser beam travels along the line $y = 3x - 2$. At the exact point where the laser crosses the y-axis, a mirror is placed perpendicular to the path of the beam.…
A laser beam travels along the line $y = 3x - 2$. At the exact point where the laser crosses the y-axis, a mirror is placed perpendicular to the path of the beam. Determine the explicit equation of the line representing the mirror.
General form and slope
Many problems give lines as $Ax + By + C = 0$. This is called the general form. It is close to the standard form $Ax + By = C$ from Integrated Math I: the only difference is that the constant $C$ sits on the left side with the opposite sign. In both forms you can read the slope straight from the coefficients.
Horizontal and vertical lines. A horizontal line has $A = 0$ (for example $y - 2 = 0$), so its slope is $0$. A vertical line has $B = 0$ (for example $x - 4 = 0$), so its slope is undefined. All horizontal lines are parallel to each other, all vertical lines are parallel to each other, and every vertical line is perpendicular to every horizontal line.
Write down the general equation of the line that is parallel to the x-axis and passes through the point $Q(-3, 8)$.
Write down the general equation of the line that is parallel to the x-axis and passes through the point $Q(-3, 8)$.
Check analytically if the lines given in general form $r: 3x - 6y + 2 = 0$ and $s: x - 2y - 5 = 0$ are parallel.
Check analytically if the lines given in general form $r: 3x - 6y + 2 = 0$ and $s: x - 2y - 5 = 0$ are parallel.
Calculate the value of the parameter $k$ so that the line $kx - 3y + 6 = 0$ is parallel to the line joining the points $A(1, 2)$ and $B(4, 11)$.
Calculate the value of the parameter $k$ so that the line $kx - 3y + 6 = 0$ is parallel to the line joining the points $A(1, 2)$ and $B(4, 11)$.
Parallel and perpendicular in general form
You do not have to solve for $y$ every time. Use the slopes $-\frac{A_1}{B_1}$ and $-\frac{A_2}{B_2}$ and simplify.
The two lines above are $x + 2y - 3 = 0$ and $2x - y + 7 = 0$. Check the test: $1\cdot 2 + 2\cdot(-1) = 0$. They are perpendicular.
Determine the value of the parameter $m$ so that the lines $r: mx + 4y - 1 = 0$ and $s: 2x - 5y + 8 = 0$ are perpendicular.
Determine the value of the parameter $m$ so that the lines $r: mx + 4y - 1 = 0$ and $s: 2x - 5y + 8 = 0$ are perpendicular.
The adjacent sides of a rectangular plot lie on the lines $r: 2x - 4y + 7 = 0$ and $s: 4x + 2y - 3 = 0$. Show analytically that the sides form a right angle. Then, find…
The adjacent sides of a rectangular plot lie on the lines $r: 2x - 4y + 7 = 0$ and $s: 4x + 2y - 3 = 0$. Show analytically that the sides form a right angle. Then, find the equation of a line parallel to $r$ that passes through the origin.
On a city map, Main Street follows the path of the line $2x + 5y - 10 = 0$. Market Street is perpendicular to it and passes through a square located at the coordinates…
On a city map, Main Street follows the path of the line $2x + 5y - 10 = 0$. Market Street is perpendicular to it and passes through a square located at the coordinates $P(-2, 3)$. Find the general equation of Market Street.
Direction vectors
A direction vector $\vec v = \langle a, b\rangle$ of a line is any nonzero vector that points along the line. You may also see it written $(a, b)$. Moving along $\vec v$ means going $a$ units right and $b$ units up, so the slope is $\dfrac{b}{a}$.
A direction vector of the line $r$ is $\vec{v} = (3, -1)$. Write down a direction vector for a line that is parallel to $r$ and another for a line that is perpendicular…
A direction vector of the line $r$ is $\vec{v} = (3, -1)$. Write down a direction vector for a line that is parallel to $r$ and another for a line that is perpendicular to $r$.
Parametric and symmetric equations
A line through $P(x_0, y_0)$ with direction vector $\langle a, b\rangle$ can be described by a moving point.
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 the first and passes through the control tower $T(0, 0)$. Write down the parametric equations of the route of the second aeroplane.
A main irrigation pipe follows the line with continuous equation $\frac{x-2}{4} = \frac{y+1}{-3}$. A secondary pipe needs to be installed perpendicular to the main one,…
A main irrigation pipe follows the line with continuous equation $\frac{x-2}{4} = \frac{y+1}{-3}$. A secondary pipe needs to be installed perpendicular to the main one, connecting to a sprinkler located at $S(-1, 5)$. Find the continuous equation of this secondary pipe.
Given the lines $r: \frac{x+1}{2} = \frac{y-3}{k}$ and $s: 3x - 5y + 2 = 0$, determine the value of $k$ so that they are parallel. Then, calculate the value of $k$ so…
Given the lines $r: \frac{x+1}{2} = \frac{y-3}{k}$ and $s: 3x - 5y + 2 = 0$, determine the value of $k$ so that they are parallel. Then, calculate the value of $k$ so that they are perpendicular.
The perpendicular bisector
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 solar panel is aligned with the line passing through the points $P(1, 4)$ and $Q(5, 2)$. To maximize efficiency, the sun's rays must strike the panel perpendicularly.…
A solar panel is aligned with the line passing through the points $P(1, 4)$ and $Q(5, 2)$. To maximize efficiency, the sun's rays must strike the panel perpendicularly. Write down the parametric equations of the path of a sunbeam that hits exactly at the midpoint of the panel.
Right angles and unknown parameters
To prove that a shape has right angles or parallel sides, compare slopes (or use the coefficient tests).
The roof design of a sports hall includes two crossing steel beams. Beam 1 rests on the line $ax + 3y - 5 = 0$ and beam 2 on the line $2x - by + 4 = 0$. Calculate the…
The roof design of a sports hall includes two crossing steel beams. Beam 1 rests on the line $ax + 3y - 5 = 0$ and beam 2 on the line $2x - by + 4 = 0$. Calculate the exact values of $a$ and $b$ knowing that the beams are perpendicular and that beam 1 passes through the point $P(1, -2)$.
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 the adjacent angles are right angles.
Two robot vacuum cleaners move around an open-plan room. The first follows the path given by the equation $y = \frac{1}{2}x + 2$. The second moves along a line…
Two robot vacuum cleaners move around an open-plan room. The first follows the path given by the equation $y = \frac{1}{2}x + 2$. The second moves along a line perpendicular to the first, such that they would collide at the exact point of intersection of the first path with the y-axis. Find the explicit equation of the second robot's path and calculate the area of the triangle formed by the paths of both robots and the x-axis.
Distance to a line, projections, and altitudes
The shortest distance from a point $P$ to a line $r$ is along the segment perpendicular to $r$. The point $H$ where that perpendicular meets $r$ is the foot of the perpendicular, also called the orthogonal projection of $P$ onto $r$.
Altitudes and the orthocenter. An altitude of a triangle is the line through a vertex perpendicular to the opposite side. The three altitudes meet at one point, the orthocenter. Find two altitudes and intersect them.
A ship sails in a straight line following the path $r: 3x + 4y - 24 = 0$. A lighthouse is located at the point $F(7, 7)$. Calculate the shortest distance from the…
A ship sails in a straight line following the path $r: 3x + 4y - 24 = 0$. A lighthouse is located at the 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 that passes through $P$).
Calculate the exact coordinates of the orthocenter (point of intersection of the three altitudes) of the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(2, -5)$.
Calculate the exact coordinates of the orthocenter (point of intersection of the three altitudes) of the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(2, -5)$.
Back to the city map: if avenues have slope $2$, then cross streets with slope $-\tfrac12$ are perpendicular to them, since $2\cdot\left(-\tfrac12\right) = -1$. A boulevard with slope $3$ is neither parallel (slope $\neq 2$) nor perpendicular (product $\neq -1$). It just cuts across at an angle.