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 $O(0, 0)$ and angle $\alpha = 90^\circ$ in the positive direction (counterclockwise). 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 line segment with endpoints $A(1, 1)$ and $B(3, 1)$. If we apply a point symmetry with center at the origin $O(0, 0)$, we know that this transformation is equivalent to a rotation through a given angle.

1. What is that angle of rotation?

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

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 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 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 center 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 center through any angle, the resulting figure is identical to the original.

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

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 center at the origin $O(0, 0)$ through an angle of $+90^\circ$ (counterclockwise). Plot the original triangle and its image $A'B'C'$, stating the coordinates of the new vertices.

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 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 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.

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 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 center 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 centered 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 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.

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 is commutative.

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

1. a) What is the minimum number of sides the regular polygon must have?
2. b) Calculate all 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 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 followed using a compass or by reasoning about the distances to the center of rotation.

A fragment of an ancient Arabic mosaic has been found 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 transformation 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 transformation. Once located, state its coordinates and calculate the magnitude 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 center $O(0, 0)$ and angle $180^\circ$, and to the result, another rotation with center $C'(-4, -3)$ (which is the image of $C$ after the first rotation) and angle $90^\circ$ counterclockwise.

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 speed camera 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 through an angle of $270^\circ$ counterclockwise relative to its own position $R$, determine the new coordinates $V'$ of the vehicle on the control center map. Use a grid of squares to perform the rotation step by step, analyzing the horizontal and vertical displacements relative to the center $R$.

Analyze 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 (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 centers 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).