Exercises: Distances in the plane

Calculate the exact distance between the points $A(3, -2)$ and $B(-1, 1)$.

Find the distance from the origin to the line of equation $3x - 4y + 10 = 0$.

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 slope-intercept form $y = 2x - 1$.

Find the distance between the parallel lines $r: 3x + 4y - 5 = 0$ and $s: 3x + 4y + 15 = 0$.

Check if 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.

Calculate the distance from the point $P(-4, 7)$ to the y-axis and the x-axis.

Given the line 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 surveying team has delimited a triangular plot of land whose base stations are located at the coordinates $A(1, 2)$, $B(5, -1)$ and $C(4, 6)$, expressed in miles. Calculate the exact area of the land by first evaluating the length of the base $AB$ and then the corresponding altitude from the vertex $C$.

A telecommunications antenna is to be installed on a straight road that coincides with the x-axis. This antenna must be exactly at the same distance from town $P(-2, 3)$ and town $Q(4, 1)$. At what exact coordinates should the antenna be located?

Two ships follow straight and 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 direction. Calculate the minimum distance between both trajectories.

In a park, the center of a square is located at the point $P(0, 5)$. A straight irrigation pipe follows the trajectory $y = x + 1$. A sprinkler is to be installed on the pipe exactly $5$ units away from the center of the square. Find the coordinates of the two possible locations for the sprinkler.

An industrial laser must make a main cut in a metal plate following 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 $10$ inches on each side of the main cut. Find the general equations of the lines of the safety cuts.

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 what exact point in its trajectory the vehicle will be at the minimum distance from the base and calculate this distance.

A solar panel has 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 feet. Calculate the exact perimeter of the panel.

A comet passes through a star system following a rectilinear trajectory of 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.

Find the possible values of the parameter $k$ so that the line $r: kx + 3y - 5 = 0$ is exactly $2$ units away from the point $P(1, -2)$.

The lines $r: x - 2y + 1 = 0$ and $s: 2x + y - 3 = 0$ represent the edges of two avenues that cross. A bike lane is to be built that is equidistant from both avenues at all times. Find the equations of the two possible trajectories of the bike lane (the angle bisectors) and prove analytically that they are perpendicular.

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.

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 to 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 is contained in the line $d: x - y + 1 = 0$.

Find the equation of the locus of the 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.

Calculate the possible values of the parameter $c$ knowing that the parallel lines $r: 2x - y + c = 0$ and $s: 4x - 2y + 5 = 0$ are $3\sqrt{5}$ units apart.

A rhombus has one of its vertices at the point $A(2, 1)$ and the diagonal that does not pass through $A$ lies on the line $r: x + 2y - 4 = 0$. If the area of the rhombus is $20$ u$^2$, find the coordinates of the remaining three vertices.