Distances in the plane
A drone flying over the Malecón in Santo Domingo, a laser cutting a metal sheet, or an antenna that must give signal to two towns all need to measure the same thing: how far a point is from another point or from a straight path. In the Cartesian plane, both questions have their own formula, and both come from the Pythagorean theorem.
Distance between two points
If you move from $A$ to $B$, you travel some amount horizontally and some amount vertically. Those two displacements are the legs of a right triangle, and the segment $AB$ is the hypotenuse.
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.
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 $A$.
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 meters. 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 meters. Calculate the exact perimeter of the panel.
Distance from a point to a line
The distance from a point to a line is the smallest of all the distances from the point to the points on the line, and it is measured along the perpendicular segment.
The idea behind the formula: $Ax_0+By_0+C$ measures how far $P$ is from satisfying the equation of the line, and dividing by $\sqrt{A^2+B^2}$ (the magnitude of the vector $(A,B)$, which is perpendicular to $r$) turns it into an actual length.
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 $3x - 4y + 10 = 0$.
Determine the distance from the point $P(2, 5)$ to the line $r: 5x + 12y - 4 = 0$.
Determine the distance from the point $P(2, 5)$ to the line $r: 5x + 12y - 4 = 0$.
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
Since the coefficients are the same, there is a shortcut: if $r: Ax+By+C=0$ and $s: Ax+By+C'=0$, then
$$d(r,s)=\frac{|C'-C|}{\sqrt{A^2+B^2}}$$Lines parallel at a given distance
Now the other way around: given $r: Ax+By+C=0$ and a distance $d$, which lines parallel to $r$ are at that distance? They have the form $Ax+By+C'=0$, and by the previous shortcut, $|C'-C|=d\sqrt{A^2+B^2}$. This gives two values of $C'$: one line on each side of $r$.
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 height of a triangle is the distance from a vertex to the line containing the opposite side. With this you can compute the area without any other formulas.
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 kilometers.…
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 kilometers. 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.
Equidistant points and minimum distances
Many problems ask for points that satisfy a distance condition. The method is always the same: call the unknown point $(x, y)$, write the condition using the distance formulas, and solve.
Equidistant from two points
Equidistant from two lines
When you remove the absolute value there are two options, $+$ and $-$, and you get two lines that are perpendicular to each other.
Minimum distance to a line
If a vehicle moves along a line, the closest point to a base $C$ is the foot of the perpendicular drawn from $C$ to the line. You find it like this: write the perpendicular line through $C$ and solve the system with the given line. The minimum distance is $d(C, r)$.
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 center 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 center 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 center 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.
Find the possible values of the parameter $k$ such that the distance from the line $r: kx + 3y - 5 = 0$ to the point $P(1, -2)$ is exactly $2$ units.
Find the possible values of the parameter $k$ such that the distance from the line $r: kx + 3y - 5 = 0$ to the point $P(1, -2)$ is exactly $2$ units.
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.
Find the coordinates of the points in the plane that simultaneously satisfy two conditions: they are equidistant from the intersecting lines $r: 3x - 4y + 2 = 0$ and…
Find the coordinates of the points in the plane that simultaneously satisfy two conditions: they are equidistant from the intersecting lines $r: 3x - 4y + 2 = 0$ and $s: 4x + 3y - 1 = 0$, and their distance from the origin is exactly $5$ units.
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.
Diagonals of squares and rhombuses
When a figure has a diagonal on a known line and a vertex off that line, distances give you the other vertices.
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 $d: x - y - 1 = 0$.
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.
Go back to the beginning: the drone, the laser, and the antenna are all solved with these same formulas, the point-to-point one and the point-to-line one.