Exercises: Rotations

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

Given the point $P(4, 2)$ in 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 $P'$?

Look at the recycling logo composed of three arrows. What is the order of this rotational symmetry? Based on that order, calculate the minimum angle required for the logo to coincide exactly with itself when rotated.

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

1. a) What is that angle of rotation?

1. b) Draw in your notebook the original segment and its image $A'B'$, and write the coordinates of $A'$ and $B'$.

You can do it on paper and send a photo with your answer.

At a pizzeria, a family-size pizza has been cut into $8$ equal slices. In the figure, the blue slice is the starting one. If you rotate the circular tray through an angle of $135^\circ$ clockwise (orange arrow), how many slice positions has each piece of pizza moved?

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

Two consecutive rotations with the same centre $O$ have been applied to a triangle: the first of $40^\circ$ and the second of $50^\circ$, both in the positive direction (counterclockwise). The figure shows the initial triangle $T_0$, the triangle $T_1$ after the first rotation and the triangle $T_2$ after the second.

1. a) What single rotation is equivalent to this composition?

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

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 centre of rotation.

• b) A rotation is an indirect isometry because it reverses the sense of angles.

• c) If we rotate a circle about its own centre through any angle, the resulting figure is identical to the original.

• d) The composition of two rotations with different centres is always another rotation with the same centre as the first one.

Cian has to rotate the quadrilateral with vertices $A(2, 3)$, $B(5, 3)$, $C(5, 1)$, and $D(2, 1)$ by an angle of $90^\circ$ counterclockwise about the origin $O(0, 0)$. On his exam, he writes: "Since the rotation is $90^\circ$ counterclockwise, the rule consists of changing the signs of both coordinates of each point, so the transformed vertex will be $A'(-2, -3)$". a) Identify and explain with reasoning the error made by Cian: to which geometric transformation does the rule he applied actually correspond? b) Write the correct algebraic rule to transform a generic point $(x, y)$ by a $90^\circ$ counterclockwise rotation about the origin. c) Apply the correct rule to calculate the true coordinates of the four transformed vertices: $A'$, $B'$, $C'$, and $D'$.

When applying a rotation about the origin $O(0, 0)$ with an angle $\alpha = 90^\circ$ counterclockwise to a figure, one of its vertices ends up at the position $P'(4, -5)$. a) Deduce with reasoning the coordinates that the original point $P$ had before performing the rotation. b) If you had wanted to move the point $P$ to exactly the same final position $P'$ by means of a rotation about the origin but in a clockwise (negative) direction, what angle should you have used? c) Verify that the distance to the centre of rotation remains unchanged by calculating the distance from $P$ to $O$ and the distance from $P'$ to $O$ using the Pythagorean theorem.

A graphic designer is creating a new logo based on a triangle with vertices $A(2, 1)$, $B(5, 1)$ and $C(4, 3)$, which you can see in the figure. To complete the design, they need to rotate this triangle about the origin of coordinates $O(0, 0)$ by an angle of $+90^\circ$ (counterclockwise). Copy the original triangle in your notebook, graph its image $A'B'C'$ and state the coordinates of the new vertices.

You can do it on paper and send a photo with your answer.

Look at the following figure representing a wind turbine blade assembly. If the assembly 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 through so that the assembly coincides exactly with its initial position.
3. c) If the assembly rotates at a constant speed and takes $2$ seconds to complete one full rotation, through what angle will it have rotated after $0.5$ seconds? Give your answer in degrees and determine whether, at that instant, the figure visually coincides with the original one.

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

1. a) What is the name of 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.

You can do it on paper and send a photo with your answer.

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 centre of rotation $O$, we have joined point $A$ of the original figure with its image $A'$ in the rotated figure, and point $B$ with its image $B'$.

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

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

1. a) Calculate the angle of rotation (in degrees) described by the minute hand after $20$ minutes. Remember to state the sign of the angle according to the direction of rotation of the clock.
2. b) If the minute hand is $10\text{ cm}$ long, what is the distance travelled by the tip of the hand in those $20$ minutes? (Treat the trajectory as a circular arc).
3. c) Through what angle will the hour hand have rotated in that same time interval?

Let there be a square centred at point $C(2, 2)$ with sides parallel to the coordinate axes and length $2$ units.

1. a) State the coordinates of its four vertices.
2. b) If we apply a rotation of $+90^\circ$ with centre at the square's own centre $C(2, 2)$, 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.

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

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

1. a) What is the minimum number of sides the regular polygon must have?

1. b) For the polygon in part a), calculate all the rotation angles greater than $0^\circ$ and less than $360^\circ$ that leave the figure invariant.

1. c) If the polygon were a regular octagon, what would be the minimum rotation angle for the figure to overlap itself?

In the air traffic control system of the Dublin Airport, the control tower is located at the origin of coordinates $O(0, 0)$. An approaching aircraft is at point $A(-3, 6)$, where each unit represents $1\text{ nautical mile}$. The tower orders it to perform a holding pattern consisting of a $270^\circ$ counterclockwise rotation centered at the tower itself. a) Before calculating the coordinates using any formula, estimate with reasoning in which quadrant of the plane the aircraft will be after completing the three-quarter turn, and predict the sign of each of its coordinates. b) Without using a calculator, estimate between which two consecutive integers the distance from the aircraft to the tower (distance $OA$) lies. c) Calculate the exact coordinates of the final position $A'$ using the algebraic rotation rule and find the exact distance from $A'$ to the tower. Compare these results with your previous estimates and assess whether they were accurate.

The Met Éireann weather radar network scans the sky by sweeping the horizon in concentric circles. Taking the radar antenna as the origin of coordinates $O(0, 0)$ (where $1\text{ unit} = 10\text{ km}$), technicians locate the core of a storm at point $T(-4, 3)$. To calibrate the sensors across different sectors, the software simulates the rotation of the weather map. a) If the software first applies a $90^\circ$ counterclockwise rotation about the origin, and then applies another $180^\circ$ rotation with the same center to the resulting position, calculate the final coordinates of the storm's core after both movements. b) Model the composition of both rotations mathematically: to what single equivalent rotation (specifying angle in degrees and direction) does consecutively applying $90^\circ$ and $180^\circ$ counterclockwise correspond? Verify that applying this single rotation to $T$ yields the same point. c) The early warning system triggers if a storm enters within a detection radius of $60\text{ km}$ from the antenna. Analytically prove whether this storm will trigger the alarm and justify why no rotation centered at the radar would change the status of said alarm.

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 centre $C(3, 1)$ and angle $\alpha = 90^\circ$ anticlockwise.

Draw the coordinate system in your exercise book, 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 followed using a compass or by reasoning about the distances to the centre of rotation.

You can do it on paper and send a photo with your answer.

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 moved 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 find the center of rotation $O$ of this movement graphically. Once you have located it, state its coordinates and calculate the measure of the angle of rotation $\alpha$ and its direction.

In a laser security system, a mirror located at point $M(0, 2)$ must rotate a beam of light. The beam originally arrives along the direction of the 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 $A$?

In the figure, each grid square measures 1 unit.

Calculate the coordinates of the new point $A'$ approximately using technical drawing, and check whether the distance to the center of rotation stays constant using the Pythagorean theorem.

Consider a square with vertices $A(2, 1)$, $B(4, 1)$, $C(4, 3)$ and $D(2, 3)$. A composite transformation is applied to it: first a rotation with centre $O(0, 0)$ and angle $180^\circ$, and to the result, another rotation with centre $C'(-4, -3)$ (which is the image of $C$ after the first rotation) and angle $90^\circ$ anticlockwise.

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

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?

A traffic radar is located at point $R(2, 2)$ on a map of the control center. A vehicle is at point $V(5, 6)$.

We rotate point $V$ through an angle of $270^\circ$ counterclockwise about center $R$. This gives us point $V'$, which is the new position of the vehicle on the map.

Determine the coordinates of $V'$. Use the square grid in the figure to carry out the rotation step by step, analyzing the horizontal and vertical displacements of $V$ with respect to the center $R$.

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

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

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

3. If a shape has two distinct centres of point symmetry, $O_1$ and $O_2$, prove that the shape must be infinite. (Hint: consider the composition of two rotations of $180^\circ$ and what transformation results).

You want to rotate point $P(5, 2)$ by an angle of $90^\circ$ counterclockwise, but the center of rotation is not the origin, but point $C(2, -1)$. Two students propose different strategies to find the coordinates of the resulting point $P'$ without drawing with a protractor:

• Laura's method (cross relative displacement): She calculates the displacement from center $C$ to $P$, checking how much to move horizontally and vertically. She reasons that, in a $90^\circ$ counterclockwise rotation, "moving right" becomes "moving up", and "moving up" becomes "moving left". She then applies these displacements starting from center $C$.

• David's method (algebraic translation to the origin): He subtracts the coordinates of $C$ from those of $P$ to fictitiously translate the center to $(0, 0)$. In that auxiliary system, he applies the known formula $(-y, x)$, and finally adds the coordinates of $C$ to return the point to its real position.

1. a) Apply Laura's method step by step and determine the coordinates of $P'$. b) Apply David's method step by step and check if you obtain exactly the same result. c) Compare both methods: analyze which is more intuitive for avoiding sign errors in manual paper-and-pencil calculation, and which would be more efficient if you had to program a computer algorithm to rotate a grid of $500$ points. Justify your conclusions.

In a digital animation, a rigid segment modeling a door handle rotates on the screen. In the initial position, the endpoints of the handle are at $A(4, 1)$ and $B(8, 1)$. After applying a $90^\circ$ counterclockwise rotation about an unknown center $C(x_0, y_0)$, the endpoints end up at positions $A'(2, 5)$ and $B'(2, 9)$. a) Show that the motion is a rigid transformation by verifying that the length of the handle is preserved ($d(A, B) = d(A', B')$). b) A designer claims that the center of rotation is the origin of coordinates $O(0, 0)$. Prove that their claim is false by checking whether the equidistance condition $OA = OA'$ holds. c) Knowing that any relative displacement $(x - x_0, y - y_0)$ from the center is transformed by a $90^\circ$ counterclockwise rotation into $(-(y - y_0), x - x_0)$, use point $A(4, 1)$ and its image $A'(2, 5)$ to set up a system of two linear equations with two unknowns ($x_0$ and $y_0$). d) Solve the system to find the exact coordinates of the center of rotation $C$ and verify that, using that same center, point $B(8, 1)$ correctly transforms into $B'(2, 9)$.

Unit Challenge. How does mathematics harness the wind to generate clean energy? You will design the rotor of a three-bladed wind turbine and analyze its motion using geometric rotations.

1. Real data collection: Access the EirGrid website (Smart Grid Dashboard) and write down the percentage of electricity demand covered by wind energy in real time today at $12:00$.

2. Design in GeoGebra: On the Cartesian plane, place the centre of rotation at $O(0, 0)$. Draw a basic blade (triangle or polygon) with a vertex at $A(0, 5)$. Generate the other two blades by applying rotations with centre $O$ and angles of $120^\circ$ and $240^\circ$ counterclockwise.

3. Simulation and coordinates: Create an angle slider $\alpha \in [0^\circ, 360^\circ]$ to rotate the entire assembly. Obtain in the application the coordinates of vertex $A$ after a rotation of $\alpha = 45^\circ$, verifying that its components are around $(-3.54; 3.54)$.

4. Final product: Write a short report with the EirGrid data, screenshots of your construction, and a conclusion: why does an angle of $120^\circ$ guarantee the mechanical balance of the wind turbine?

Assessment criteria:

• Rigour in the geometric definition of rotation (centre, angle, and direction) in GeoGebra.

• Correct retrieval of real wind generation data from EirGrid.

• Precision in reading and calculating rotated coordinates with decimal points.

• Quality of the technical and geometric justification in the report.