Rotations

A graphic designer is creating a pattern and applies a rotation of $270^\circ$ in the positive direction to a figure. His colleague says that the same result would have been obtained by applying a single rotation in the negative (clockwise) direction. What angle should that negative rotation be for the figures to coincide?

Given point $P(4, 2)$ on the Cartesian plane, perform a rotation with center at the origin of coordinates $O(0, 0)$ and angle $\alpha = 90^\circ$ in the positive (counterclockwise) direction. What are the coordinates of the transformed point $P'$?

A graphic designer is creating a new logo based on a triangle with vertices $A(2, 1)$, $B(5, 1)$ and $C(4, 3)$. To complete the design, they need to rotate this triangle with its center at the origin $O(0, 0)$ and an angle of $+90^\circ$ (counterclockwise). Graphically represent the original triangle and its image $A'B'C'$, stating the coordinates of the new vertices.

Consider a square centered at point $C(2, 2)$ with sides parallel to the coordinate axes and of length $2$ units.

1. a) State the coordinates of its four vertices.
2. b) If we apply a rotation of $+90^\circ$ with center at the square's own center $C(2, 2)$, what are the new coordinates of the vertices?
3. c) Are there any points of the figure that have not changed position after the rotation? Justify your answer.

A video game designer is programming the movement of a magic key on a Cartesian plane. The key is defined by a line segment joining the points $A(1, 1)$ and $B(1, 4)$. The program must perform a rotation with center $C(3, 1)$ and angle $\alpha = 90^\circ$ counterclockwise.

Draw the coordinate system in your notebook, plot the points and the original figure, and determine the coordinates of the new points $A'$ and $B'$ after the rotation. Explain the procedure you have followed using a compass or by reasoning about the distances to the center of rotation.

A traffic radar located at point $R(2, 2)$ rotates sweeping a circular sector. An auxiliary camera located at the origin $O(0, 0)$ captures an image of a vehicle at position $V(5, 6)$.

If the radar rotates the vehicle by an angle of $270^\circ$ counterclockwise with respect to its own position $R$, determine the new coordinates $V'$ of the vehicle on the control center map. Use a square grid to perform the rotation step by step, analysing the horizontal and vertical displacements with respect to the center $R$.

In a laser security system, a mirror located at point $M(0, 2)$ must rotate a light beam. The beam originally arrives following the direction of the line segment joining $A(-4, 2)$ to $M$. If we apply a rotation with center $M$ and angle $\alpha = -45^\circ$ (clockwise), what will be the new position of the point that was at $A$?

Calculate the coordinates of the new point $A'$ approximately using technical drawing and check if the distance to the center of rotation remains constant using Pythagoras' theorem.

We have a line segment with endpoints $A(1, 1)$ and $B(3, 1)$. If we apply a central symmetry with center at the origin $O(0, 0)$, we know that this transformation is equivalent to a rotation by a certain angle.

1. What is that angle of rotation?

2. Draw the original segment and its transformed segment $A'B'$.

In your notebook, draw a line segment $PQ$ with coordinates $P(1, 4)$ and $Q(3, 2)$. Apply a rotation to the segment with center $C(1, 1)$ and angle $\alpha = 180^\circ$.

1. a) What name is given to this special type of rotation when the angle is $180^\circ$?
2. b) Find the coordinates of the transformed points $P'$ and $Q'$.
3. c) Check by calculating distances whether the length of the segment has been preserved.

We have two figures in the plane, $F$ and $F'$. We know that $F'$ is the figure obtained after applying a rotation to $F$. To find the center of rotation $O$, we have joined point $A$ of the original figure with its corresponding point $A'$ in the rotated figure, and point $B$ with its corresponding point $B'$.

Explain in detail the geometric procedure you must follow using the perpendicular bisector of the segments $AA'$ and $BB'$ to locate the center of rotation. What would happen if the segments $AA'$ and $BB'$ were parallel?

We have found a fragment of an ancient Arabic mosaic where a triangle with vertices $P(2, 2)$, $Q(5, 2)$ and $R(4, 4)$ has been displaced by an unknown rotation. We know that the image of point $P$ after the movement is $P'(8, 2)$ and the image of $Q$ is $Q'(8, 5)$.

Use the properties of perpendicular bisectors to graphically find the center of rotation $O$ of this movement. Once located, state its coordinates and calculate the size of the angle of rotation $\alpha$ and its direction.

Two consecutive rotations have been applied to a triangle with the same center $O$: the first of $40^\circ$ and the second of $50^\circ$, both in the positive direction.

1. What single rotation is this composition equivalent to?

2. If we now apply a third rotation so that the triangle returns to its original position, what angle and direction must that rotation be?

We perform a composition of two transformations on a point $P(3, -1)$. First we apply a rotation with center $O(0, 0)$ and angle $\alpha = +90^\circ$ obtaining $P'$. Next, we apply to $P'$ another rotation with the same center $O$ and angle $\beta = +180^\circ$ obtaining $P''$.

1. a) Determine the final coordinates of point $P''$.
2. b) Is there a single rotation that allows us to go directly from $P$ to $P''$? If so, state its center and angle.
3. c) Would the same result be obtained if we applied the rotation of $180^\circ$ first and then that of $90^\circ$? Justify whether the composition of rotations with the same center satisfies the commutative property.

Consider a square with vertices $A(2, 1)$, $B(4, 1)$, $C(4, 3)$ and $D(2, 3)$. A composition of movements is applied to it: first a rotation with center $O(0, 0)$ and angle $180^\circ$, and to the result, another rotation with center $C'(-4, -3)$ (which is the corresponding point of $C$ after the first rotation) and angle $90^\circ$ counterclockwise.

1. Determine the coordinates of the vertices after the first movement.

2. Determine the final coordinates after the composition.

3. Is there a single rotation that produces the same result? If so, what would its angle be?

Show graphically in your notebook that the composition of two reflections in two axes $r$ and $s$ that intersect at a point $O$ forming an angle of $30^\circ$ is equivalent to a rotation.

To do this:

1. Draw any point $P$ outside the axes.

2. Find $P'$, its reflection in $r$.

3. Find $P''$, the reflection of $P'$ in $s$.

4. Measure the angle $\widehat{POP''}$. What is its relationship with the angle formed by the axes? State the general property that relates the angle of the axes to the angle of the resulting rotation.

A ceiling fan has $5$ blades distributed completely symmetrically. If the fan is switched off and you push it gently causing one of the blades to move to the exact position where the next one was, what angle has the fan rotated? State the angle in both the positive (counterclockwise) and negative (clockwise) direction.

Look at the recycling logo made up of three arrows. What is the order of this center of rotation? From that order, calculate the minimum angle needed for the logo to coincide exactly with itself when rotated.

In a pizzeria, they have cut a family pizza into $8$ equal slices. If you rotate the circular tray by an angle of $135^\circ$ in a clockwise direction, how many slice positions has each piece of pizza moved?

In an Arabic mosaic, a minimum motif is used which is repeated by rotations. If the center of a regular polygon of $n$ sides is the center of rotation, and we know that when rotating the figure by an angle of $40^\circ$ it does not coincide with the original, but when rotating it $72^\circ$ it does:

1. a) What is the minimum number of sides the regular polygon must have?
2. b) Calculate all the angles of rotation less than $360^\circ$ that leave the figure invariant.
3. c) If the polygon were a regular octagon, what would be the minimum angle of rotation for the figure to map onto itself?

A circular logo has a design that is repeated by rotations. If the design coincides with itself for the first time when rotated $40^\circ$ about its center:

1. What is the order of rotational symmetry of this figure?

2. State all the angles of rotation less than $360^\circ$ that leave the figure invariant.

3. If the logo also had axes of symmetry, how many axes would it have at a minimum? Justify your answer based on the relationship between the order of rotational symmetry and the axes of symmetry in regular polygons and rosettes.

Look at the following figure which represents a wind turbine blade. If the blade is a polygon with rotational symmetry, determine:

1. a) The order of rotational symmetry of the figure.
2. b) The minimum angle it must rotate so that the blade coincides exactly with its initial position.
3. c) If the blade rotates at a constant speed and takes $2$ seconds to complete a full rotation, what angle will it have rotated after $0,5$ seconds? Express the result in degrees and determine if at that moment the figure visually coincides with the original.

A wall clock shows exactly $12:00$ o'clock.

1. a) Calculate the angle of rotation (in degrees) described by the minute hand when $20$ minutes pass. Remember to indicate the sign of the angle according to the direction of rotation of the clock.
2. b) If the minute hand measures $10\text{ cm}$, what is the distance traveled by the tip of the hand in those $20$ minutes? (Consider the path as an arc of a circle).
3. c) What angle will the hour hand have rotated in that same time interval?

State which of the following statements about rotations in the plane are true and which are false. Briefly justify your answer:

• a) In a rotation of $180^\circ$, the only point that does not move is the center of rotation.

• b) A rotation is an indirect isometry because it changes the orientation of angles.

• c) If we rotate a circle around its own center by any angle, the resulting figure is identical to the original.

• d) The composition of two rotations with a different center is always another rotation with the same center as the first.

Analyse the invariant elements of a rotation in the plane. If we apply a rotation with center $C(x, y)$ and an angle $\alpha \neq 0^\circ$ and $\alpha \neq 360^\circ$:

1. How many invariant points (that do not change position) exist in the whole plane?

2. Are there invariant lines (that are transformed into themselves even if their points change position)? Study the special case where the angle of rotation is $180^\circ$.

3. If a figure has two different centers of point reflection, $O_1$ and $O_2$, show that the figure must be infinite. (Hint: consider the composition of two rotations of $180^\circ$ and what movement results).