Distances in the plane
A drone, a laser cutter or a radar system constantly need to measure the same thing: how far one point is from another point, or from a straight path. In the Cartesian plane, both questions have their own formula, and both are derived from Pythagoras.
Distance between two points
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.
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 $3x - 4y + 10 = 0$.
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 $y = 2x - 1$.
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.
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 using determinant formulae: 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.
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?
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.