Translations

Identify which of the following options correctly describes a translation:

1. a) Turning a key in a lock.
2. b) The movement of a drawer when opening it.
3. c) The reflection of a mountain in a lake.
4. d) The movement of the hands of a clock.

Look at the grid and determine the components of the translation vector $\vec{u}$ that allows moving the point $P$ to the position of the point $P'$.

In a video game, a character is at point $C(5, -2)$. The player presses a key combination that is equivalent to a translation. If the character suddenly appears at point $C'(1, 3)$, what is the translation vector $\vec{w}$ that has acted?

A surveillance drone takes off from a control point and makes a linear movement defined by the vector $\vec{v} = (5, -3)$. If the starting point has coordinates $A(2, 4)$ on the control center map, what will be the coordinates of the point $A'$ where the drone is located after making this movement?

If we translate the point $Q$ by the vector $\vec{v} = (-4, 7)$ we obtain the point $Q'(0, 2)$. What were the coordinates of the original point $Q$ before the translation?

In a graphic design application, a teenager is creating a logo. They have drawn a triangle with vertices at $P(2, -1)$, $Q(5, 0)$ and $R(3, 4)$. To complete the design, they need to translate the triangle so that vertex $P$ is situated at the origin $O(0, 0)$.

1. a) State the components of the translation vector $\vec{v}$ that must be applied.
2. b) Calculate the new coordinates of the vertices $Q'$ and $R'$ after applying this translation.

Apply a translation by vector $\vec{v} = (6, 2)$ to the triangle with vertices $V_1(0, 0)$, $V_2(2, 0)$ and $V_3(1, 3)$.

1. Write the coordinates of the new vertices.

2. How does the area of the new triangle compare to the original? Justify your answer.

Consider the quadrilateral $ABCD$ whose vertices are $A(1, 1)$, $B(4, 1)$, $C(5, 3)$ and $D(2, 3)$.

1. a) Apply a translation by vector $\vec{t} = (3, -4)$ and write the coordinates of the image figure $A'B'C'D'$.
2. b) Calculate the area of the original quadrilateral and check if the translation has modified its value. Justify your answer based on the properties of isometries.

Given the points $A(-2, 3)$ and $B(4, 7)$, a translation $T_1$ by vector $\vec{u} = (k, 2k)$ is applied to them. After this movement, the point $A$ is transformed into $A'$. If we know that the modulus of the vector $\vec{AA'}$ is $10$ units and that $k > 0$:

1. Find the value of $k$.

2. Calculate the coordinates of $A'$ and $B'$.

3. Show that the line segment $A'B'$ has the same length as the line segment $AB$. What property of translations is demonstrated by this?

On the Cartesian plane, we have a line $s$ that passes through the points $A(0, 2)$ and $B(2, 6)$. We apply a translation by vector $\vec{v} = (1, 3)$ to this line.

1. a) Calculate the coordinates of the transformed points $A'$ and $B'$.
2. b) Find the slope of the original line $s$ and of the translated line $s'$. What relationship do you observe between them?
3. c) Are there any translation vectors that leave the line $s$ invariant (that is, the transformed line coincides with the original)? Give an example of one of these vectors.

Consider the line $r$ whose equation is $y = 2x - 3$. We know that a translation leaves a line invariant (that is, the line is transformed into itself) if the translation vector is parallel to that line.

1. Find a translation vector $\vec{u} = (a, b)$ with modulus $\sqrt{20}$ that leaves the line $r$ invariant.

2. If we apply that same vector $\vec{u}$ to the point $P(1, -1)$, which belongs to the line, what are the coordinates of the transformed point $P'$? Check analytically if $P'$ still belongs to the line $r$.

On a Cartesian plane, a figure is first translated by the vector $\vec{u} = (2, -3)$ and then by an unknown vector $\vec{v} = (v_x, v_y)$. The resulting translation (composition) is a translation that leaves all the points of the y-axis ($Y$-axis) now situated on the line $x = 5$.

1. Determine the composite vector $\vec{w} = \vec{u} + \vec{v}$.

2. Find the components of the vector $\vec{v}$.

3. If we apply the inverse translation of $\vec{v}$ to the point $Q(0, 0)$, at what point is it situated?

Given the circle with center $C(3, -2)$ and radius $r = 4$ units:

1. a) Find the equation or the coordinates of the center $C'$ of the circle translated by the vector $\vec{u} = (-5, 6)$.
2. b) What is the radius of the new circle?
3. c) If a point $P(7, -2)$ belongs to the original circle, calculate the coordinates of its corresponding point $P'$ after the translation.

In the design of a video game, a circular character with center at $C(x, y)$ and radius $R = 3$ must dodge an obstacle. The initial equation of its outline is $(x+4)^2 + (y-2)^2 = 9$.

The character is translated by the composition of two vectors: $\vec{u} = (7, 3)$ and $\vec{v} = (-2, 5)$.

1. Determine the coordinates of the center $C$ before and after the composite translation.

2. Write the equation of the circle in its final position.

3. If the obstacle is a vertical wall located on the line $x = 5$, explain with reasoning whether the character collides with the wall after making the movement.

An object undergoes two consecutive translations. The first is defined by the vector $\vec{u} = (2, 5)$ and the second by the vector $\vec{v} = (3, -1)$. What is the vector of the single translation that is equivalent to performing the two previous ones in succession?

A surveillance drone takes off from a base located at point $A(-4, 1)$ on a coordinate system. To carry out its first patrol, it moves following the translation vector $\vec{u} = (6, 3)$. After a pause, it makes a second movement defined by the vector $\vec{v} = (-2, 5)$.

1. a) Calculate the coordinates of the final point where the drone is located after the two movements.
2. b) Determine the components of a single vector $\vec{w}$ that represents the direct displacement from the take-off point to the final position.

A video game character moves across the screen following a straight-line path. The programmer has defined its movement through the composition of three successive translations:

1. $\vec{v_1} = (5, -2)$

2. $\vec{v_2} = (-3, 7)$

3. $\vec{v_3} = (a, b)$

If the character starts at point $(0, 0)$ and after the three movements ends at point $(4, 4)$, what must the components of the third vector $\vec{v_3}$ be?

Two translations are said to be inverse if, when applied one after the other, the final point coincides with the initial point (the identity is obtained).

1. a) If an object has been translated by the vector $\vec{v} = (-7, 4)$, what is the vector of the inverse translation?
2. b) If we apply the translation $\vec{u} = (3, -2)$ to the point $M(x, y)$ and then its inverse, what are the coordinates of the resulting point?
3. c) A boat moves from the port $A(10, 20)$ to a buoy $B$ by means of the vector $\vec{v} = (50, -30)$. Due to a storm, it must return immediately to the port. What translation vector must it follow to return exactly to the point $A$?

A movement in the plane is defined as the composition of three successive translations defined by the vectors $\vec{a} = (3, -1)$, $\vec{b} = (-5, 4)$ and $\vec{c} = (2, -3)$.

1. Show analytically that this composition of movements is equivalent to the identity translation.

2. What does it mean geometrically for the result to be the identity?

3. If a square with an area of $16$ square units undergoes these three translations, what will its final area be? Justify your answer based on the properties of isometries.

A surveillance drone departs from a base located at point $B(12, 15)$. It makes a first displacement following the vector $\vec{u} = (-8, 4)$ to position itself over a target. After completing the mission, it receives an emergency displacement order via the vector $\vec{v} = (5, -10)$.

1. Calculate the final coordinates of the drone after both movements.

2. Determine the single translation vector $\vec{w}$ that would allow the drone to return directly from its final position to the base $B$ in a single movement.

3. If each unit on the grid represents $10$ feet, calculate the total distance (modulus) it would travel on that return journey.

A team of architects is designing a frieze for a new middle school. The basic motif is a letter "L" with vertices at $(0,0), (0,3), (1,3), (1,1), (2,1)$ and $(2,0)$.

1. a) Draw the motif on graph paper.
2. b) To generate the frieze, they must successively apply the translation by vector $\vec{w} = (3, 0)$. Write the coordinates of the vertices of the second and third motifs of the frieze.
3. c) What would be the translation vector needed to go directly from the first motif to the fifth?

An architect designs a frieze by repeating a polygonal motif. The vertices of the basic motif are $A(0, 0)$, $B(2, 0)$, $C(3, 2)$ and $D(1, 2)$. To generate the frieze, they successively apply the translation by vector $\vec{t} = (4, 0)$.

1. Calculate the coordinates of the vertices of the third motif of the frieze (the one obtained after applying the translation twice consecutively to the original motif).

2. If we want the frieze to be a composition of two different translations, $\vec{t_1}$ and $\vec{t_2}$, such that $\vec{t_1} + \vec{t_2} = (4, 0)$ and the first translation $\vec{t_1}$ displaces the motif upwards by $2$ units, what must the vector $\vec{t_1}$ and the vector $\vec{t_2}$ be?

An equilateral triangle has its vertices at the points $A(0, \sqrt{3})$, $B(-1, 0)$ and $C(1, 0)$. It undergoes a translation that maps the centroid of the triangle to the origin $O(0, 0)$.

Note: The centroid of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is calculated as $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$.

1. Calculate the coordinates of the original centroid $G$.

2. Determine the necessary translation vector $\vec{v}$.

3. Calculate the new coordinates of the vertices $A', B'$ and $C'$.