Translations

Look at the grid, in which each side of a small square measures one unit, and determine the components of the translation vector $\vec{u}$ that moves point $P$ to the position of point $P'$.

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.

Given the line segment with endpoints $A(-3, 1)$ and $B(1, 2)$, calculate the coordinates of the transformed segment $A'B'$ after applying a translation with vector $\vec{v} = (4, 3)$. Plot both line segments on a set of coordinate axes.

Apply a translation with 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 one? Give reasons for your answer.

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

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

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 vertices $Q'$ and $R'$ after applying this translation.

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

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

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

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

In the Cartesian plane, we have a line $s$ passing through 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 gradient of the original line $s$ and of the translated line $s'$. What relationship do you notice between them?
3. c) Are there translation vectors that leave line $s$ invariant (that is, such that the transformed line coincides with the original)? Give an example of one such vector.

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

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

2. If we apply this same vector $\vec{u}$ to point $P(1, -1)$, which lies on the line, what are the coordinates of the transformed point $P'$? Verify analytically whether $P'$ still lies on line $r$.

Given 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, point $A$ is transformed into $A'$. If we know that the magnitude of 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. Prove that line segment $A'B'$ has the same length as line segment $AB$. What property of translations is demonstrated by this?

An equilateral triangle has its vertices at 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 required translation vector $\vec{v}$.

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

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 equivalent to carrying out the previous two one after the other?

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

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

A warehouse robot moves along the floor of the building following a straight path. The technician has defined its movement using the composition of three consecutive translations:

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

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

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

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

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

3. If each grid unit represents $10$ meters, calculate the total distance (magnitude) it would cover on that return trip.

In video game design, 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 moves via the composition of two vectors: $\vec{u} = (7, 3)$ and $\vec{v} = (-2, 5)$.

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

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

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

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 repeatedly apply 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 $\vec{t_1}$ shifts the motif upwards by $2$ units, what should vector $\vec{t_1}$ and vector $\vec{t_2}$ be?

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

1. Prove analytically that this composition of transformations is equivalent to the identity translation.

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

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