4. Rotations of Figures in a Plane
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Chapter 1
4.
Rotations of Figures in a Plane
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Why
Rotations in geometry involve turning a figure about a fixed point, known as the center of rotation. This concept is crucial in understanding how shapes and figures transform within a plane. The center of rotation plays a pivotal role, as every point of the figure rotates around this center at a specific angle, leading to a new position for each point. Tools like straightedges, compasses, and protractors can aid in manually performing these rotations. Real-life examples of rotations can be observed in everyday objects like doors, where the hinges act as the center of rotation. Additionally, the human body, especially joints like elbows, showcases rotational movements. This topic delves deeper into the relationships between a point and its image under rotation, providing a comprehensive understanding of the concept.
Lesson Settings & Tools
| | 15 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Rotations of Figures in a Plane
Slide of 15
When learning about transformations, rotations were mentioned and used briefly. In this lesson, rotations will be studied more deeply by analyzing the relationships between a point Q and its image Q' under a rotation.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Rotating Points About a Center
In the following applet, points A, B, C, and D can be moved and rotated about point P.
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Explore
Rotating a Triangle About a Center
Perform a rotation to △ JKL about point P. Use a protractor to find the measure of ∠ JPJ', ∠ KPK', and ∠ LPL'.
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Discussion
Rotations Along Circles
The first exploration shows the preimage and the image of a point under a rotation are the same distance from the center of rotation. That is, the image moves along a circle passing through the preimage and centered at the center of rotation.
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Discussion
Defining a Rotation
Identifying whether one figure is the image of another figure under rotation can be difficult. A key aspect to observe is whether the center of rotation is the same distance from an image as it is from its preimage.
Concept
Rotation of Geometric Objects
A rotation is a transformation in which a figure is turned about a fixed point P. The number of degrees the figure rotates α ^(∘) is the angle of rotation. The fixed point P is called the center of rotation. Rotations map every point A in the plane to its image A' such that one of the following statements is satisfied.
- If A is the center of rotation, then A and A' are the same point.
- If A is not the center of rotation, then A and A' are equidistant from P, with ∠ APA' measuring α ^(∘).
Rotations are usually performed counterclockwise unless stated otherwise.
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Example
Identifying Rotations
Consider the following three triangles and a point P. Use the given measuring tool to find the distance from each vertex to P and the angles formed by each preimage, the point P, and the corresponding image.
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Hint
Remember, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. The angle of rotation is formed by a preimage, the center of rotation, and the corresponding image.
Solution
Remember that, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. Then, start by finding the distances between each vertex and P.
Notice that the vertices of △ A'B'C' are further from P than the vertices of △ ABC. Consequently, △ A'B'C' cannot be the image of △ ABC under a rotation about P. Next, find and compare the measures of ∠ APA', ∠ BPB', and ∠ CPC'.
As can be seen, ∠ APA', ∠ BPB', and ∠ CPC' have all the same measure, which is 150^(∘) when measured counterclockwise or 210^(∘) when measured clockwise. Therefore, △ A''B''C'' is the image of △ ABC under a rotation about P. The angle of rotation is either 150^(∘) or 210^(∘).
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Discussion
Rotations Performed by Hand
Rotations can be performed by hand with the help of a straightedge, a compass, and a protractor.
To rotate point A about point P by an angle of 130^(∘) measured counterclockwise, follow these five steps.
1
Draw PA
expand_more 
2
Place the Protractor
expand_more Place the center of the protractor on P and align it with PA.
The protractor is placed as illustrated above when the rotation is counterclockwise. If the rotation has to be done clockwise, the protractor needs to be placed as follows.
3
Mark the Desired Angle
expand_more Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at 130^(∘).
4
Draw a Ray
expand_more Using the straightedge, draw a ray with starting point P that passes through the mark made in the previous step.
5
Draw Point A'
expand_more Place the compass tip on P and open it to the distance between P and A. Without changing this setting and keeping the point of the compass at P, draw a small arc centered at P that intersects the ray drawn before.
The intersection of the ray and the arc is the image A' after the give rotation.
Notice that this method of construction has also confirmed that PA is congruent to PA'.
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Example
Rotating a Triangle
On a geometry test, Ignacio was asked to perform a 70^(∘) counterclockwise rotation to △ ABC about point P.
Draw △ ABC and its image under this rotation.
Answer
Solution
To rotate △ ABC, the rotation can be performed on each vertex, one at a time. For example, start by rotating A. To do so, first draw PA using a straightedge.
Then, place the center of the protractor on P and align it with PA in such a way that the rotation is counterclockwise. Then, make a small mark at 70^(∘).
Next, draw a ray with starting point P that passes through the mark made before.
Finally, place the compass tip on P and open the compass to the distance between P and A. With this setting, make an arc that intersects the ray. The intersection point is the image of A under the rotation.
Vertices B and C can be rotated following the same steps.
Finally, the image of △ ABC under the given rotation is the triangle formed by A', B', and C'.
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Pop Quiz
Practice Rotations
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Discussion
Rotating Points by 90^(∘), 180^(∘), and 270^(∘)
In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a counterclockwise rotation about the origin. This relation occurs when the angle of rotation is either 90^(∘), 180^(∘), or 270^(∘). Try to figure it out by using the following applet.
- The image of (a,b) under a 90^(∘) rotation about the origin is (- b,a).
- The image of (a,b) under a 180^(∘) rotation about the origin is (- a,- b).
- The image of (a,b) under a 270^(∘) rotation about the origin is ( b,- a).
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Example
Finding the Center of Rotation
Given a figure and its image under a rotation, the following theorem can be used to find the center of rotation.
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If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the segment. |
With this theorem in mind, consider the following example. In the diagram, quadrilateral A'B'C'D' is the image of ABCD under a certain rotation.
Find the center and angle of rotation.
Answer
Angle of Rotation: 120^(∘) clockwise or 240^(∘) counterclockwise.
Graph:
Graph:
Hint
Remember that the center of rotation is equidistant from the preimage and the image of each vertex. Use the Converse Perpendicular Bisector Theorem. The center is the intersection point between two perpendicular bisectors.
Solution
The first step is to find the center of rotation. Remember, by definition, a point and its image under a rotation are the same distance from the center.
The center of rotation is equidistant from a point and its image.
Therefore, by the Converse of the Perpendicular Bisector Theorem, the center lies on the perpendicular bisector of AA', for instance. Then, with the aid of a compass and a straightedge, start by constructing the perpendicular bisector of this segment.
To determine the center's exact position, draw a second segment joining a vertex and its image, for example, DD'. Then, draw the perpendicular bisector of this segment. The intersection between both perpendicular bisectors is the center of rotation.
Notice that drawing only two perpendicular bisectors is enough to find the center of rotation because all will intersect at the same point. Since the sense of rotation was not specified, both measures will be found using a protractor.
The angle of rotation is either 240^(∘) counterclockwise or 120^(∘) clockwise.
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Pop Quiz
Finding the Center and Angle of a Rotation
In the following applet, the solid triangle is the image of the dashed triangle under a certain counterclockwise rotation. Place the point where the center of the rotation should be. Then, select the appropriate angle of rotation.
Extra
About the AppletIf the point is not placed close enough to the center of rotation, when the
Check Answerbutton is pushed, a red area is highlighted indicating the region where the center of rotation is located.
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Example
Performing a Composition of Rotations
Recall that rotations are transformations and that transformations can be composed. Therefore, it is possible to have a composition of two or more rotations. On a geometry exercise, the following two rotations are given.
- R_1: 90^(∘) counterclockwise rotation about C_1(0,0).
- R_2: 180^(∘) counterclockwise rotation about C_2(0,-1).
LaShay has to perform both rotations to △ ABC, one after the other, but the book does not indicate the composition's order.
According to the book, the correct coordinates of B'' are (2,-1). Knowing this, help LaShay to find the coordinates of C'' and draw △ A''B''C''.
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Answer
Solution
Start by applying any of the given two rotations. For example, apply R_1 first. Notice it is a 90^(∘) counterclockwise rotation about the origin. Then, it maps (a,b) onto (- b,a). Knowing this, the image of A, B, and C can be quickly found.
ccc
A(-4,1) & → & A'(-1,-4) B(-1,2) & → & B'(-2,-1) C(-3,4) & → & C'(-4,-3)
Therefore, the image of △ ABC under R_1 looks as follows.
Next, apply R_2 to △ A'B'C'. Remember, it is a 180^(∘) counterclockwise rotation about C_2(0,-1).
Since the coordinates of B'' are the same as the book says, the rotations were applied in the correct order. Consequently, the coordinates of C'' are (4,1). Notice that applying the rotations in the reverse order would produce a different image.
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Discussion
Composition of Rotations About Different Centers
Think about the preimage and the final image obtained by LaShay in the previous example. Is it possible to map △ ABC onto △ A''B''C'' performing only one rotation? The answer is yes! A 270^(∘) counterclockwise rotation about (-1,-1) does it.
- Place the centers A and B at any place within the square formed by the axes and the point (-1,1).
- The slider on the left performs a rotation about point A and the slider on the right performs a rotation about point B.
- Perform rotations such that the sum of their angles of rotation equals 360^(∘).
- Compare the preimage and the image.
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Closure
Rotations in Real Life
In real life, there are plenty of situations where rotations can be appreciated. For instance, take a look at a door.
- To open a door, the handle must be rotated.
- To lock or unlock it, a key must be inserted and rotated.
- While the door is opening, it rotates around the hinges.
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Rotations of Figures in a Plane
Exercise 3.1
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The statement tells us that as long as the distances to a given point and what would be corresponding vertices in two shapes are congruent, this must describe a rotation about that point.
General Rotation
Let's show an example of a general triangle ABC that has undergone a rotation about a point P.
As we can see, the rotation fulfills the given criterion regarding segment length between corresponding vertices and the center of rotation. PA&=PA' && ✓ PB&=PB' && ✓ PC&=PC' && ✓
Is This Always True?
Remember, a rotation is a rigid motion. Therefore, if it is possible to change at least one of the triangle's vertices, while keeping these three equations true, then the given statement that this describes a rotation is false. Below we have moved A' in a way that keeps PA=PA' true.
Since we can draw two triangles that fulfills the criterion without △ A'B'C' being a rotation of △ ABC, the statement must be false.
This time, we are given that the angle of rotation between what would be corresponding vertices on a rotated triangle, its preimage, and a point P, are all congruent. This is indeed a criterion when performing a rotation.
As we can see, the given angle criterion is fulfilled. m∠ APA'= m∠ BPB'=m∠ CPC' ✓ However, there is another criterion for a rotation. In addition to every angle between corresponding vertices and the center of rotation being congruent, the segments connecting P with corresponding vertices must also be congruent. Below we have moved one of the image's vertices without changing the angle of rotation.
As we can see, the given angle criterion is still fulfilled. m∠ APA'= m∠ BPB'=m∠ CPC' ✓ However, this shows a different transformation than △ ABC. Therefore, the statement is false.
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Hint & Answer
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Solution
Point A has been rotated 120^(∘) counterclockwise about a center of rotation P until it ends up at B.
Find the coordinates of P.
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When a polygon is rotated, the coordinates of the center of rotation P can be found by following these steps.
- Connect at least two pairs of corresponding vertices.
- Construct the perpendicular bisectors of these segments.
- Find the point of intersection of the perpendicular bisectors.
Below we see an example of this.
However, in this case, we are rotating a point. This means that we can only determine the perpendicular bisector of one segment.
The center of rotation must fall somewhere on this perpendicular bisector. But where? Remember that the center of rotation is equidistant to the image and preimage. This means the center of rotation, along with the points A and B, form an isosceles triangle ABP. Recall the Isosceles Triangle Theorem.
Isosceles Triangle Theorem |- If two sides of a triangle are congruent, then the angles opposite them are congruent.
Additionally, because the angle of rotation is the vertex angle of this isosceles triangle, we can calculate its base angles by using the Interior Angles Theorem. Let's label these base angles x.
If we use a protractor, we can find the segments AP and BP. We have two options here. The center of rotation could be above or below AB.
Since A is rotated counterclockwise, the only viable option is P_1. We see that this point falls on (4,1).
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Solution
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