Distances in the Plane
A delivery drone in Guadalajara, a laser cutter, or a radio antenna always need to measure the same thing: how far one point is from another point, or from a straight path. On the coordinate plane both questions have a formula, and both come from the Pythagorean theorem.
Distance Between Two Points
If you move from $A$ to $B$, you travel some horizontal amount and some vertical amount. Those two displacements are the legs of a right triangle, and the segment $AB$ is the hypotenuse.
Midpoint of a Segment
The midpoint $M$ of $AB$ is halfway along both horizontally and vertically, so its coordinates are the averages of the coordinates of the endpoints:
$$M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$$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 minimum distance: the length of the segment perpendicular to the line that reaches the point.
The numerator measures how far $P$ is from satisfying the equation; the denominator $\sqrt{A^2+B^2}$ is the magnitude of the normal vector $(A, B)$ and turns that value 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 $y = 2x - 1$.
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.
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 and Rhombuses
The point-to-line distance lets you calculate the area of a triangle given by its vertices: the base is the length of one side, and the height is the distance from the opposite vertex to the line containing that side.
Area of a Rhombus
The diagonals of a rhombus are perpendicular and cross at their midpoint. They split it into four congruent right triangles, and that gives the formula:
$$\text{Area} = \frac{D \times d}{2}$$where $D$ and $d$ are the lengths of the diagonals. Also, the center of the rhombus is the midpoint of both diagonals.
To find the vertex opposite $A$, use the center $M$ (the foot of the perpendicular from $A$ to $r$): the opposite vertex is the reflection of $A$ across $M$. The other two vertices are on $r$, at a distance $D/2$ from $M$.
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.
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
Many problems ask for a point that meets a distance condition: being the same distance from two points, from two lines, or as close as possible to a line.
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.
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.
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.
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.
Back to the beginning: the drone, the laser, and the antenna solve their problem with these same formulas: Pythagoras for point to point and the perpendicular for point to line.