Before you start, make sure you know: Loci (3rd Year, Junior Cycle)
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
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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.
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.
A direction vector $(a, b)$ of a line points along it: move $a$ across and $b$ up. So the slope is $m = \dfrac{b}{a}$. Any multiple of $(a, b)$ is also a direction vector of the same line, and so of every parallel line. A perpendicular line has direction vector $(-b, a)$ (or $(b, -a)$), because its slope $\dfrac{a}{-b} = -\dfrac{1}{m}$.
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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 $r$.
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.
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 map: two lines are parallel if their slopes are equal and perpendicular if the product of their slopes is $-1$.
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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 $P$).
To write the equation of a line you need its slope and one point on it. This is in the Formulae and Tables booklet.
Parametric and symmetric forms
The line through $(x_0, y_0)$ with direction vector $(a, b)$ has parametric equations $x = x_0 + at$, $y = y_0 + bt$. Eliminating $t$ gives the symmetric form $\dfrac{x-x_0}{a} = \dfrac{y-y_0}{b}$.
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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.
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. Next, 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. Next, calculate the value of $k$ so that they are perpendicular.
A solar panel is aligned with the line passing through the points $P(1, 4)$ and $Q(5, 2)$. To maximise 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 maximise efficiency, the sun's rays must strike the panel perpendicularly. Write the parametric equations of the path of a sunbeam that strikes exactly at the midpoint of the panel.