Exercises: Geometric transformations

You are editing a photograph on your cell phone. If you apply a transformation that maintains the angles of the image and the proportion between its sides, but changes its overall size to fit into a smaller frame, what type of geometric transformation are you performing? Reason whether it is an isometry or a similarity.

A video game character is at point $A(2, 3)$. To dodge an obstacle, the player moves it following a translation vector $\vec{v} = (4, -2)$. Calculate the coordinates of the final point $A'$ where the character will stop.

Look at the following capital letters of the alphabet: A, M, T, U.

1. What type of symmetry do they have with respect to a vertical axis passing through their center?

2. Draw the letter B in your notebook and its reflection with respect to a vertical line situated to its right. Is it still the letter B or has its orientation changed?

In a graphic design studio, a logo has been created in the shape of a regular hexagon.

1. What is the minimum angle we must rotate the hexagon about its center so that the figure coincides exactly with its initial position?

2. If we rotate the figure $180^\circ$, would we get the same result as applying a point reflection with respect to the center of the hexagon?

Given the triangle with vertices $P(1, 1)$, $Q(3, 1)$ and $R(2, 4)$, apply a translation by vector $\vec{u} = (-3, 2)$. State the coordinates of the new vertices $P'$, $Q'$ and $R'$. Has the area of the triangle changed after the movement?

Identify which of the following movements is an indirect isometry and briefly explain why:

• A translation of $5\text{ cm}$ to the left.

• A rotation of $90^\circ$ clockwise.

• A reflection in the y-axis.

Draw the point $C(4, -3)$ on a Cartesian coordinate system.

1. Find its reflection in the $X$-axis (x-axis) and call it $C'$.

2. Find the reflection of $C$ in the origin $(0, 0)$ and call it $C''$.

State the coordinates of both points.

A frieze is constructed by repeating a basic motif along a straight line. If the motif is always displaced by the same distance in the same direction, what geometric transformation is being applied repeatedly? If, in addition to being displaced, the motif is reflected as in a mirror with respect to the ground line, what is this combination called?

A photo editing app allows you to apply filters that modify images. Consider a triangle with vertices $A(1, 2)$, $B(4, 2)$ and $C(2, 5)$. Indicate what type of geometric transformation has been applied in each case and whether it is an isometry:

1. a) The "Mirror" filter transforms vertex $A(1, 2)$ into $A'(-1, 2)$, $B(4, 2)$ into $B'(-4, 2)$ and $C(2, 5)$ into $C'(-2, 5)$.
2. b) The "Zoom" filter transforms the vertices by multiplying both coordinates by $2$.
3. c) The "Move" filter moves the figure $3$ units to the right and $1$ downward.

Explain the difference between a direct isometry and an inverse isometry using the letter L and the letter F as an example.

1. a) What transformation changes the letter "L" into another "L" located in a different position but with the same orientation?
2. b) Is it possible to superimpose a figure onto its image after an axial symmetry without lifting it from the plane? Reason whether this makes it a direct or an inverse isometry.

An architecture studio creates a model of a shopping center at a scale of $1:200$. The floor of the model is a quadrilateral with an area of $150 \text{ in}^2$.

1. a) Calculate the scale factor $k$ between reality and the model.
2. b) What will the real area of the shopping center floor be in square feet? (Remember the relationship between the scale factor and the area ratio).

Given the translation by vector $\vec{v} = (5, -2)$, calculate the coordinates of the vertices of the transformed triangle $A'B'C'$ if the original vertices are $A(0, 4)$, $B(-2, 1)$ and $C(3, 0)$. Graphically represent the vector that joins each point with its corresponding image.

A logo designer is working with a square with center $C(2, 2)$ and vertices at points $(1, 1)$, $(3, 1)$, $(3, 3)$ and $(1, 3)$.

1. a) Apply a rotation with center at the origin $O(0, 0)$ and angle $90^\circ$ counterclockwise. Indicate the new coordinates.
2. b) Which point of the figure remains invariant if the rotation is performed with center at point $(2, 2)$?

Draw a coordinate axis system and the line segment with endpoints $P(2, 5)$ and $Q(6, 5)$.

1. a) Find the symmetric segment $P'Q'$ with respect to the y-axis ($Y$ axis).
2. b) Find the symmetric segment $P''Q''$ with respect to the origin of coordinates $(0, 0)$.
3. c) What is the relationship between the coordinates of $P$ and $P''$? What rotation is this central symmetry equivalent to?

Two consecutive translations are applied to a point on the plane. The first translation has the vector $\vec{u} = (-3, 4)$ and the second has the vector $\vec{v} = (7, 1)$.

1. a) Calculate the resulting translation vector $\vec{w} = \vec{u} + \vec{v}$.
2. b) If the final point after the two transformations is $P''(5, 8)$, what were the coordinates of the initial point $P$?

Look at the following regular mosaic formed by hexagons.

1. a) Identify a center of rotation and the minimum angle necessary for the mosaic to coincide with itself.
2. b) Indicate how many axes of axial symmetry one of the individual hexagons has and whether the center of the hexagon is also a center of central symmetry.

In a graphic design studio, a logo is being created based on the composition of symmetries. Consider two parallel lines $r$ and $s$. Line $r$ has the equation $x = 2$ and line $s$ has the equation $x = 5$. If we first apply an axial symmetry (reflection) with respect to $r$ and, to the result, an axial symmetry with respect to $s$ on a point $P(1, 4)$:

1. a) Determine the coordinates of the final point $P''$ after both transformations.
2. b) Reasonably identify which single transformation is equivalent to this composition of symmetries, indicating its characteristic elements (vector, angle or center, as appropriate).
3. c) Would the result be the same if we first applied the symmetry with respect to $s$ and then with respect to $r$? Justify your answer by calculating the coordinates in this new order.

A handmade kaleidoscope uses two mirrors that intersect at a point $O(0, 0)$. The first mirror coincides with the x-axis ($X$) and the second mirror forms an angle of $30^\circ$ with the first. A small colored crystal is located at point $A(4, 2)$.

1. a) If a light ray is reflected first on the $X$ axis and then on the second mirror, what single transformation describes the passage of the original crystal $A$ to its second image $A''$?
2. b) Calculate the angle of rotation and the center of this transformation.
3. c) If we want the composition of the two symmetries to result in a rotation of $180^\circ$ (central symmetry), what angle should the two mirrors form with each other?

A translation by vector $\vec{u}(4, -1)$ followed by a rotation with center $O(0, 0)$ and angle $90^\circ$ counterclockwise is applied to a triangle with vertices $A(1, 2)$, $B(3, 2)$ and $C(2, 4)$.

1. a) Find the coordinates of the vertices of the final triangle $A''B''C''$.
2. b) Is the composition of transformations commutative in this case? Check this by calculating the position of vertex $A$ if the rotation were applied first and then the translation.
3. c) Describe a line that is invariant (that transforms into itself) under the translation by vector $\vec{u}$.

In a video game, a character is at point $M(-2, 3)$. The programmer needs the character to perform a "glide reflection". To do this, they must apply an axial symmetry with respect to the line $y = 1$ and then a translation by vector $\vec{v}(5, 0)$.

1. a) Find the final coordinates of the character after the movement.
2. b) Explain why this movement leaves no invariant point on the plane.
3. c) If the translation were by vector $\vec{w}(0, 5)$ instead of $\vec{v}(5, 0)$, would it still be a glide reflection? Reason your answer based on the relationship between the vector and the axis of symmetry.

An Alhambra mosaic is formed by pieces obtained through the composition of rotations. A basic piece has a center of rotation at point $C(2, 2)$. A rotation of $120^\circ$ is applied to it, followed by another rotation of $240^\circ$ with the same center $C$.

1. a) What is the final position of a point $P(4, 2)$ after both rotations?
2. b) Generalize the result: What transformation is equivalent to the composition of two rotations with the same center and angles $\alpha$ and $\beta$?
3. c) If we apply two rotations of $90^\circ$ but with different centers, $C_1(0, 0)$ and $C_2(4, 0)$, is the result another rotation or a translation? Justify your answer by analyzing the total angle.

An architect uses an enlargement (homothety) to scale the plan of a circular square. The center of the square on the original plan is at $O(0, 0)$ and a bench is located at $B(3, 4)$. An enlargement with center $C(1, 1)$ and scale factor $k = -2$ is applied.

1. a) Calculate the new coordinates of the bench $B'$ after the enlargement.
2. b) If the area of the original square was $25 \pi$ $cm^2$ on the plan, what will the area of the square be after the transformation?
3. c) What has happened to the orientation of the figure since the scale factor is negative? Is it still a direct or an inverse transformation?

Two towns, $A$ and $B$, are located on the same side of a straight road $r$. A telecommunications technician must install an antenna at a point $P$ on the road so that the sum of the distances from the towns to the antenna ($AP + PB$) is minimum.

1. a) Describe how to use axial symmetry to find the exact location of point $P$.
2. b) If in a coordinate system the towns are at $A(2, 5)$ and $B(10, 3)$, and the road is the $X$ axis, calculate the coordinates of point $P$.
3. c) Prove, based on the properties of symmetry, that for any other point $Q$ on the road, $AQ + QB > AP + PB$.

Consider a square with vertices $V_1(0, 2)$, $V_2(2, 2)$, $V_3(2, 0)$ and $V_4(0, 0)$. A transformation consisting of a central symmetry with respect to point $(1, 1)$ followed by a rotation of $180^\circ$ with center at the same point $(1, 1)$ is applied to it.

1. a) Determine the position of the vertices after the first transformation.
2. b) Determine the position of the vertices after the second transformation.
3. c) What is the resulting transformation of composing two central symmetries with different centers $C_1$ and $C_2$? Investigate this case by testing with $C_1(1, 1)$ and $C_2(3, 1)$ on vertex $V_4$. What translation vector do you obtain?