Exercises: Transformations in the plane

A delivery drone is at position $P(3, -2)$, expressed in kilometres relative to its base. The controller sends it an instruction to make a straight-line displacement defined by the translation vector $\vec{v} = (-5, 4)$. Determine the coordinates of the drone's new position.

A laser cutting machine moves in a straight line on a metal sheet from the initial point $A(-1, 5)$ to the final point $B(4, 2)$. Calculate the translation vector that describes this movement exactly.

In the design of a symmetrical garden, a fountain is to be placed at point $F'$, which is the reflection of a statue located at $E(4, 7)$ with respect to a central post located at the origin $O(0, 0)$. Find the coordinates where the fountain should be installed.

A fishing boat undergoes two consecutive movements on the open sea. First, it translates according to the vector $\vec{u} = (2, 6)$ and then according to the vector $\vec{v} = (3, -1)$. If it departed from a port located at coordinates $P(-4, -5)$, determine its final position after both displacements.

Two communications satellites, $S_1$ and $S_2$, orbit symmetrically with respect to a central space station. If $S_1$ is at position $(8, -3)$ and the space station is at $(2, 1)$, calculate the coordinates of the position of satellite $S_2$.

An architect designs a bridge whose arch has a critical anchoring point at $A(-3, 5)$. To maintain visual symmetry in a 3D rendering with respect to the water surface, which coincides with the x-axis ($y=0$), they need to locate the virtual reflection of this point under the water. Determine the coordinates of this symmetrical point.

A crane moves a straight beam horizontally and vertically, whose initial endpoints are $A(1, 1)$ and $B(5, 3)$. The movement is carried out by means of a translation defined by the vector $\vec{t} = (0, -4)$. Calculate the coordinates of the new endpoints $A'$ and $B'$ of the beam.

The trajectory of a conveyor belt follows the line with equation $y = 2x - 1$. Due to space requirements in the factory, the entire belt is translated $3$ units to the left and $2$ units upwards. Write the general equation of the new line that defines the translated trajectory.

Lucas is programming a video game and needs to calculate the position of the reflection of a character located at point $A(4, -1)$ through a central symmetry with respect to the control point $M(1, 2)$. In his notebook, he writes the following:

"Since $M$ is the midpoint between $A$ and its symmetric point $A'$, I calculate the coordinates of $A'$ by averaging $A$ and $M$:

$$A' = \frac{A + M}{2} = \left(\frac{4+1}{2},\ \frac{-1+2}{2}\right) = (2,5;\ 0,5)$$"

Identify the conceptual error that Lucas makes, explain why his result contradicts the geometric definition of central symmetry, and determine the correct coordinates of the symmetric point $A'$.

A topographic control point is located at $P(-3, 5)$ and an axial symmetry with respect to the horizontal line $r \equiv y = 1$ is applied to it.

1) Before carrying out any analytical calculation, make a reasoned estimate of which quadrant of the plane the transformed point $P'$ will lie in. Argue which coordinate will remain unchanged and which will change, mentally estimating its new value from the distance between $P$ and the axis.

2) Analytically calculate the coordinates of $P'$ using the midpoint and perpendicularity conditions, and check whether the result matches your initial estimate.

In a graphic design program, a polygon with a key vertex at point $V(5, -2)$ has been drawn. An axial symmetry (reflection) across the angle bisector of the first and third quadrants (line $y=x$) is applied. Analytically determine the coordinates of the reflected vertex $V'$.

A mobile phone mast provides coverage in a circular area whose Cartesian equation is $x^2 + y^2 - 4x + 6y - 12 = 0$. Due to interference issues with a nearby building, the company decides to translate the mast (the centre of the circle) according to the vector $\vec{v} = (-3, 5)$. Determine the general equation of the new coverage area.

A laser beam is emitted from a projector situated at point $P(-2, 4)$. In front of it, there is a plane mirror placed exactly along the line with equation $r \equiv x - 2y + 5 = 0$. Calculate the coordinates of the virtual point $P'$ (the reflection of $P$ across the mirror) from which the reflected light appears to originate.

In a video game animation, the geometric centre of a character located at $A(1, 3)$ undergoes a translation by vector $\vec{t} = (4, -2)$ followed immediately by a point symmetry (point reflection) with respect to point $C(2, 0)$. Find the final position of the character and verify analytically whether the result would be the same by applying the point symmetry first and then the translation.

A railway track runs along the line $r \equiv 3x + 4y - 12 = 0$. A town planner wants to layout a new avenue $s$ that is perfectly symmetrical to the railway track with respect to a central monument located at $C(1, -1)$. Find the general equation of the avenue $s$.

The reflector of a car headlight follows the profile of the parabola $y = x^2 - 6x + 11$. To facilitate the programming of the CNC machining tool, the engineer needs to translate the parabola so that its vertex coincides exactly with the origin. Determine the required translation vector and the new equation of the parabola.

Two forest watchtowers are located at points $T_1(-3, 1)$ and $T_2(5, 5)$. A straight firebreak is to be laid out as an axis of symmetry between both towers, such that any point on the firebreak is equidistant from $T_1$ and $T_2$. Calculate the general equation of the line defining this firebreak.

An agricultural plot is in the shape of a triangle with vertices at $A(0, 0)$, $B(4, 0)$ and $C(2, 5)$. The local council compulsory purchases the plot and, in exchange, gives the owner another plot obtained by applying a reflection across the vertical line $x = 6$ to the original triangle. Calculate the vertices of the new plot and prove analytically that the area of the plot has not changed.

Two straight and parallel rails in a railway section are described by the lines $r \equiv 2x - y + 3 = 0$ and $s \equiv 2x - y - 7 = 0$.

1) Determine the general equation of the line that acts as the axis of axial symmetry transforming rail $r$ into rail $s$.

2) If, instead of a symmetry, you wish to go from line $r$ to line $s$ by means of a translation with vector $\vec{v}$, there are infinitely many possible vectors. Determine the components and the magnitude of the translation vector $\vec{v}$ that has the minimum possible geometric length.

In an automated logistics warehouse in Zaragoza, goods move on a coordinate plane measured in metres. A package located at position $A(1, 2)$ undergoes a translation defined by the vector $\vec{v} = (6, -2)$, reaching the intermediate position $A'$. Next, the guidance system applies a central symmetry to it with respect to a control post located at $M(5, 3)$, placing it at its final destination $A''$.

1) Calculate the coordinates of positions $A'$ and $A''$.

2) If the warehouse operator had reversed the order of the instructions (first applying the central symmetry with respect to $M$ to $A$, and then the translation with vector $\vec{v}$), would the package have ended up at the same position $A''$? Justify your answer by calculating the new coordinates and stating the theoretical property that governs commutativity in the composition of rigid motions.

A ray of light travels along the line $r \equiv 2x - y + 4 = 0$ and hits a plane mirror located on the line $s \equiv x + y - 1 = 0$. Determine the general equation of the line $r'$ containing the reflected ray, knowing that this line is the reflection of $r$ across $s$.

Two villages, $A(1, 5)$ and $B(7, 2)$, need to connect to a main water pipe running in a straight line along the x-axis ($y=0$). A pumping station $P$ is to be built on the main pipe such that the total length of the secondary pipes ($AP + PB$) is minimised. Using the concept of reflection, calculate the exact coordinates of point $P$ and the minimum total length of the pipe.

Prove analytically that the composition of two reflections across parallel lines is equivalent to a translation. To do this, consider the lines $r \equiv x - 2y = 0$ and $s \equiv x - 2y + 10 = 0$. Take a generic point $P(x_0, y_0)$, find its reflection $P'$ across $r$, and then the reflection $P''$ of $P'$ across $s$. Determine the translation vector $\vec{v} = \vec{PP''}$ and check that it is perpendicular to the lines and that its magnitude is twice the distance between them.

The orbit of a comet is projected onto an astronomical plane according to the equation of the ellipse $4x^2 + 9y^2 - 24x + 36y + 36 = 0$. By translating the axes, it is possible to express this equation in its canonical form centred at the origin. Calculate the translation vector that moves the geometric centre of the conic to the origin, and write the equation of the ellipse after this translation.

The outline of a circular gear has the equation $x^2 + y^2 - 2x - 4y - 4 = 0$. This gear is reflected in an inclined mirror represented by the line $r \equiv 3x - 4y + 10 = 0$. Determine the general equation of the circle representing the reflected image of the gear.

On a billiard table, a cue ball is located at point $B(2, 3)$ and a red ball at $R(8, 6)$. A player wishes to hit the red ball by having the cue ball bounce first off the top cushion of the table, which lies along the line $y = 10$. Assuming the angle of incidence is equal to the angle of reflection, calculate the coordinates of the exact point on the cushion where the cue ball must strike to achieve its objective.

Given the parallel lines $r \equiv 2x + y - 3 = 0$ and $s \equiv 2x + y + 7 = 0$, find the translation vector $\vec{v}$ of minimum magnitude that transforms the line $r$ into the line $s$. Prove analytically that this vector is perpendicular to both lines and that its magnitude coincides with the shortest distance between them.

An isometry in the plane transforms the point $A(1, 2)$ into $A'(5, 4)$ and the point $B(3, -1)$ into $B'(7, 1)$.

1. a) Show that the distance between $A$ and $B$ is the same as that between $A'$ and $B'$.
2. b) Determine whether the described transformation corresponds to a pure translation or a point reflection, justifying your answer analytically and finding the corresponding translation vector or centre of symmetry.

Given the line $r \equiv 2x - y + 1 = 0$ and the axis of axial symmetry $s \equiv x + y - 4 = 0$, find the general equation of the symmetric line $r'$ with respect to $s$. Two students debate which procedure to follow:

• Candela proposes: "I calculate the intersection point $I = r \cap s$, which is invariant because it belongs to the axis ($I' = I$). Then I choose an auxiliary point $P \in r$ with $P \neq I$, calculate its symmetric point $P'$ with respect to $s$, and the desired line will be the one passing through $I$ and $P'$".

• Mateo proposes: "I take two distinct arbitrary points $A$ and $B$ on the line $r$, calculate the symmetric point of each ($A'$ and $B'$) by drawing perpendiculars to $s$ and finding their feet, and then I write the line joining $A'$ and $B'$".

1) Compare the computational workload and geometric advantages of both methods, giving a reasoned justification of which of the two is more efficient.

2) Apply the chosen method to obtain the general equation of the symmetric line $r'$.

In a greenhouse farm in El Ejido (Almería), a water intake $B$ is planned to be installed along the bank of a straight irrigation canal, whose path follows the line $r \equiv x - 2y + 2 = 0$. The intake must supply two crop sectors located at $A(1, 4)$ and $C(7, 6)$ (coordinates expressed in hundreds of metres). It is desired to locate $B$ on the canal in such a way that the total length of pipe installed, $L = d(A, B) + d(B, C)$, is minimal.

1) Justify theoretically, using the properties of axial symmetry as an isometry that preserves distances, why the optimal position of $B$ coincides with the intersection of the canal $r$ and the line segment joining the symmetric point of $A$ with respect to $r$ (denoted $A'$) with point $C$.

2) Analytically determine the coordinates of the symmetric point $A'$ with respect to line $r$.

3) Find the exact coordinates of the water intake $B$ on the canal that minimizes the length of the installation.

Topic challenge. How does actual wind affect a drone delivery mission in your municipality? Design and optimize a flight route by integrating official meteorological data and GeoGebra.

1. Real data: Go to the AEMET website, check the wind forecast for your town (speed in $\text{km/h}$ and direction), and define the wind vector $\vec{v}$, obtaining its Cartesian components, magnitude, and direction.

2. Routes and combinatorics: In GeoGebra, place the takeoff base $O(0,0)$ and $4$ delivery points with coordinates of your choice. Calculate how many closed delivery routes (visiting all four points without repetition) are possible using combinatorics ($P_4$). Choose and justify one of them.

3. Motions in the plane: Model the journey using successive translations (displacement vectors). Assume constant wind and choose a realistic magnitude for the drone's velocity relative to the air, $\vec{v}_{drone}$. On each leg, apply vector composition $\vec{v}_{ground} = \vec{v}_{drone} + \vec{v}$, with $\vec{v}_{ground}$ parallel to the leg, to calculate the corrected heading that compensates for drift and the time for each leg, and check that the resulting net displacement $\vec{R}$ of the closed route is zero.

4. Final report: Write a brief document (one page) with the link or screenshot of your GeoGebra construction, the record of algebraic calculations, and a conclusion on the feasibility of the route.

Assessment criteria:

• Rigour in calculating components, magnitudes, directions, and the combinatorial count.

• Accuracy when modeling translations and vector addition using real AEMET data in GeoGebra.

• Technical quality, clarity, and justification in the report's conclusions.