Distances in the plane
On a Luas map of Dublin, the distance between the Heuston and Connolly stops and how far a new bus stop is from a straight stretch of the Liffey quays are the same kind of question: how far is one point from another point, or from a straight path? In the Cartesian plane, both questions have their own formula, and both are derived from Pythagoras. We will come back to the Luas at the end.
Distance between two points
Given the segment with endpoints $A(2, 4)$ and $B(6, -2)$, first calculate the coordinates of its midpoint $M$, and then find the distance from $M$ to the endpoint $A$.
Given the segment with endpoints $A(2, 4)$ and $B(6, -2)$, first calculate the coordinates of its midpoint $M$, and then find the distance from $M$ to the endpoint $A$.
Calculate the exact distance between the points $A(3, -2)$ and $B(-1, 1)$.
Calculate the exact distance between the points $A(3, -2)$ and $B(-1, 1)$.
Calculate the distance from the point $P(-4, 7)$ to the y-axis and to the x-axis.
Calculate the distance from the point $P(-4, 7)$ to the y-axis and to the x-axis.
Check whether the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 3)$ is isosceles by calculating the exact length of its three sides.
Check whether the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 3)$ is isosceles by calculating the exact length of its three sides.
A solar panel is in the shape of a quadrilateral with vertices at the coordinates $A(-1, -1)$, $B(3, 0)$, $C(4, 4)$ and $D(0, 3)$, measured in metres. Calculate the…
A solar panel is in the shape of a quadrilateral with vertices at the coordinates $A(-1, -1)$, $B(3, 0)$, $C(4, 4)$ and $D(0, 3)$, measured in metres. Calculate the exact perimetre of the panel.
Distance from a point to a line
Find the distance from the origin to the line with equation $3x - 4y + 10 = 0$.
Find the distance from the origin to the line with equation $3x - 4y + 10 = 0$.
Calculate the distance from the point $Q(-1, 3)$ to the line given in explicit form $y = 2x - 1$.
Calculate the distance from the point $Q(-1, 3)$ to the line given in explicit form $y = 2x - 1$.
A comet passes through a star system following a straight line trajectory with equation $3x + y - 6 = 0$. An observation satellite is stationary at the coordinates…
A comet passes through a star system following a straight line trajectory with equation $3x + y - 6 = 0$. An observation satellite is stationary at the coordinates $S(2, 4)$. Calculate the minimum distance at which the comet will pass the satellite.
Distance between parallel lines
Find the distance separating the parallel lines $r: 3x + 4y - 5 = 0$ and $s: 3x + 4y + 15 = 0$.
Find the distance separating the parallel lines $r: 3x + 4y - 5 = 0$ and $s: 3x + 4y + 15 = 0$.
Two ships follow straight, parallel routes. The first ship sails along the line $r: x - 2y + 4 = 0$. The second ship passes through the point $A(3, 1)$ maintaining the…
Two ships follow straight, parallel routes. The first ship sails along the line $r: x - 2y + 4 = 0$. The second ship passes through the point $A(3, 1)$ maintaining the same heading. Calculate the minimum distance separating the two paths.
An industrial laser needs to make a main cut in a metal sheet along the line $r: 4x - 3y + 2 = 0$. In addition, the system must be programmed to make two parallel safety…
An industrial laser needs to make a main cut in a metal sheet along the line $r: 4x - 3y + 2 = 0$. In addition, the system must be programmed to make two parallel safety cuts at an exact distance of $10$ mm on each side of the main cut. Find the general equations of the lines for the safety cuts.
Calculate the possible values of the parameter $c$, knowing that the distance between the parallel lines $r: 2x - y + c = 0$ and $s: 4x - 2y + 5 = 0$ is $3\sqrt{5}$…
Calculate the possible values of the parameter $c$, knowing that the distance between the parallel lines $r: 2x - y + c = 0$ and $s: 4x - 2y + 5 = 0$ is $3\sqrt{5}$ units.
Areas of triangles using base and height
The distance from a point to a line allows you to calculate the area of a triangle without needing the coordinate area formula: the base is the length of one side and the height is the distance from the opposite vertex to the line containing that side.
A surveying team has marked out a triangular plot of land whose base stations are located at coordinates $A(1, 2)$, $B(5, -1)$ and $C(4, 6)$, expressed in kilometres.…
A surveying team has marked out a triangular plot of land whose base stations are located at coordinates $A(1, 2)$, $B(5, -1)$ and $C(4, 6)$, expressed in kilometres. Calculate the exact area of the land, first evaluating the length of the base $AB$ and then the corresponding height from the vertex $C$.
An isosceles triangle has its unequal side (the base) on the line $r: x + y - 2 = 0$. The vertex opposite the base is $A(4, 4)$ and it is known that the area of the…
An isosceles triangle has its unequal side (the base) on the line $r: x + y - 2 = 0$. The vertex opposite the base is $A(4, 4)$ and it is known that the area of the triangle is $12$ u$^2$. Calculate the exact length of the base and of the two equal sides.
A rhombus has one of its vertices at point $A(2, 1)$, and the diagonal that does not pass through $A$ lies on the line $r: x + 2y - 9 = 0$. If the area of the rhombus is…
A rhombus has one of its vertices at point $A(2, 1)$, and the diagonal that does not pass through $A$ lies on the line $r: x + 2y - 9 = 0$. If the area of the rhombus is $20$ u$^2$, find the coordinates of the remaining three vertices.
Equidistant points and minimum distances
A telecommunications antenna is to be installed on a straight road that coincides with the x-axis. This antenna must be equidistant from village $P(-2, 3)$ and village…
A telecommunications antenna is to be installed on a straight road that coincides with the x-axis. This antenna must be equidistant from village $P(-2, 3)$ and village $Q(4, 1)$. What exact coordinates should the antenna be placed at?
In a park, the centre of a square is located at the point $P(0, 5)$. A straight irrigation pipe follows the path $y = x + 1$. A sprinkler is to be installed on the pipe…
In a park, the centre of a square is located at the point $P(0, 5)$. A straight irrigation pipe follows the path $y = x + 1$. A sprinkler is to be installed on the pipe at a distance of exactly $5$ units from the centre of the square. Find the coordinates of the two possible locations of the sprinkler.
A space exploration vehicle moves in a straight line following the equation $y = 2x$. An emergency wireless charging base is located at the point $C(5, 0)$. Determine at…
A space exploration vehicle moves in a straight line following the equation $y = 2x$. An emergency wireless charging base is located at the point $C(5, 0)$. Determine at which exact point on its trajectory the vehicle will be at the minimum distance from the base, and calculate that distance.
The lines $r: x - 2y + 1 = 0$ and $s: 2x + y - 3 = 0$ represent the edges of two intersecting avenues. A cycle lane is to be built that is always equidistant from both…
The lines $r: x - 2y + 1 = 0$ and $s: 2x + y - 3 = 0$ represent the edges of two intersecting avenues. A cycle lane is to be built that is always equidistant from both avenues. Find the equations of the two possible paths for the cycle lane (the angle bisectors) and show analytically that they are perpendicular.
Determine the coordinates of the vertices of a square knowing that one of its vertices is $A(1, 2)$ and that one of its diagonals lies on the line $d: x - y - 1 = 0$.
Determine the coordinates of the vertices of a square knowing that one of its vertices is $A(1, 2)$ and that one of its diagonals lies on the line $d: x - y - 1 = 0$.
Find the equation of the locus of points in the plane $P(x, y)$ whose distance to the line $r: 3x - 4y = 0$ is exactly twice their distance to the fixed point $Q(0, 2)$.…
Find the equation of the locus of points in the plane $P(x, y)$ whose distance to the line $r: 3x - 4y = 0$ is exactly twice their distance to the fixed point $Q(0, 2)$. Simplify the resulting equation.