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 with 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 explicit form $y = 2x - 1$.

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

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.

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

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 $A$.

Cian has to calculate the distance from the point $P(-2, 3)$ to the line $r: 4x - 3y + 2 = 0$. On his exam, he writes the following procedure:

1. a) Identify and explain with reasoning the two serious conceptual errors made by Cian in his solution.
2. b) Correct the exercise by carrying out the proper calculation and express the exact distance with a rationalized denominator.

Consider the point $P(2, 4)$ and the line $r: 3x - 4y + 2 = 0$.

1. a) Without applying the point-to-line distance formula yet, calculate the coordinates of the point $Q \in r$ that has the same abscissa as $P$ (that is, with $x = 2$). Find the vertical distance between $P$ and $Q$, and reason geometrically why you know with total certainty that $d(P, r) \le d(P, Q)$, with this value serving as an upper bound for the actual distance.
2. b) Calculate the exact value of $d(P, r)$ using the general formula and check whether your estimate and bound from the previous section were correct and consistent.

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 $C$.

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?

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.

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.

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.

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 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.

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.

A technical unit of Adif monitors a straight section of railway track that, on a topographic grid with coordinates in kilometers, follows the line $r: 3x - 4y + 12 = 0$. A telecommunications base station has been installed at point $T(6, 2)$.

1. a) Model the situation by calculating the exact minimum distance in kilometers between the base station and the railway track.
2. b) The railway safety protocol establishes an exclusion zone of $2\text{ km}$ on each side of the track axis in which there cannot be equipment without electromagnetic shielding. Determine with reasoning whether station $T$ requires such shielding or if it is located outside the zone.
3. c) From station $T$, it is planned to launch an inspection drone with a maximum radio link range of $5\text{ km}$. Justify geometrically whether the drone can reach some point on the railway track and reason how many points on the track are located at exactly $5\text{ km}$ from the base.

Find the general equations of the two lines parallel to $r: 5x - 12y + 8 = 0$ that are at a distance of exactly $3$ units from the point $A(1, -1)$. Next, determine the distance separating both parallel lines from each other and explain how this last value can be deduced immediately by symmetry, without needing to apply any additional formula.

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 avenues. Find the equations of the two possible paths for the cycle lane (the angle bisectors) and show 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 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 $d: x - y - 1 = 0$.

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.

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.

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.

To calculate the distance between the two parallel lines $r: 3x - 4y + 7 = 0$ and $s: 6x - 8y - 11 = 0$, three working methods are proposed:

• Method A (Definition): Take an arbitrary point $P \in r$ and calculate $d(P, s)$ using the point-to-line distance formula.

• Method B (Orthogonal projection): Draw a common perpendicular line to both, find the intersection points $M \in r$ and $N \in s$, and calculate the magnitude of the vector $\vec{MN}$.

• Method C (Simplified formula): Multiply the equation of $r$ by $2$ to match the coefficients $A$ and $B$ with those of $s$, and directly apply the expression $d(r, s) = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.

1. a) Apply Method A and Method C, solving both completely, and verify that they yield exactly the same result.
2. b) Prove algebraically the validity of the formula from Method C: take a generic line $r: Ax + By + C_1 = 0$, choose on it a generic point $P(x_0, y_0)$ that satisfies $Ax_0 + By_0 = -C_1$, and calculate $d(P, s)$ with respect to $s: Ax + By + C_2 = 0$.
3. c) Critically compare the three methods: indicate which is the most efficient in practice, which carries a greater risk of error due to the accumulation of calculations, and why Method B turns out to be unnecessarily laborious for this purpose.

Consider the intersecting lines $r: 4x - 3y + 6 = 0$ and $s: 3x + 4y - 12 = 0$. Points located on the x-axis ($OX$) that are simultaneously equidistant from both lines are sought.

1. a) Write the coordinates of a generic point on the x-axis, state the algebraic condition $d(P, r) = d(P, s)$, and obtain the exact coordinates of the only two points, $P_1$ and $P_2$, that satisfy this equality.
2. b) Determine the general equations of the two angle bisectors, $b_1$ and $b_2$, of the angles formed by $r$ and $s$. Verify by direct substitution that $P_1$ belongs to one of them and $P_2$ to the other.
3. c) Prove that the two angle bisectors $b_1$ and $b_2$ are perpendicular to each other and reason what geometric meaning the fact that the points found in section a) are precisely the intersections of each angle bisector with the x-axis has.