rotations common core geometry homework answers

Related Topics: Lesson Plans and Worksheets for Geometry Lesson Plans and Worksheets for all Grades More Lessons for Geometry Common Core For Geometry

New York State Common Core Math Geometry, Module 1, Lesson 13

Worksheets for Geometry

Student Outcomes

• Students manipulate rotations by each parameter—center of rotation, angle of rotation, and a point under the rotation.

Exploratory Challenge

You need a pair of scissors and a ruler.

Cut out the 75° angle on the right and use it as a guide to rotate the figure below 75° counterclockwise around the given center of rotation (Point 𝑃).

• Place the vertex of the 75° angle at point 𝑃.
• Line up one ray of the 75° angle with vertex 𝐴 on the figure. Carefully measure the length from point 𝑃 to vertex 𝐴.
• Measure that same distance along the other ray of the reference angle, and mark the location of your new point, 𝐴′.
• Repeat these steps for each vertex of the figure, labeling the new vertices as you find them.
• Connect the six segments that form the sides of your rotated image.

In Grade 8, we spent time developing an understanding of what happens in the application of a rotation by participating in hands-on lessons. Now, we can define rotation precisely. First, we need to talk about the direction of the rotation. If you stand up and spin in place, you can either spin to your left or spin to your right. This spinning to your left or right can be rephrased using what we know about analog clocks: spinning to your left is spinning in a counterclockwise direction, and spinning to your right is spinning in a clockwise direction. We need to have the same sort of notion for rotating figures in the plane. It turns out that there is a way to always choose a counterclockwise half-plane for any ray: The counterclockwise half-plane of 𝐶𝑃 is the half-plane of 𝐶𝑃 that lies to the left as you move along 𝐶𝑃 in the direction from 𝐶 to 𝑃. (The clockwise half-plane is then the half-plane that lies to the right as you move along 𝐶𝑃 in the direction from 𝐶 to 𝑃.) We use this idea to state the definition of rotation.

For 0° < 𝜃° < 180°, the rotation of 𝜃 degrees around the center 𝐶 is the transformation 𝑅𝐶,𝜃 of the plane defined as follows:

• For the center point 𝐶, 𝑅𝐶,𝜃 (𝐶) = 𝐶, and
• For any other point 𝑃, 𝑅𝐶,𝜃 (𝑃) is the point 𝑄 that lies in the counterclockwise half-plane of 𝐶𝑃 , such that 𝐶𝑄 = 𝐶𝑃 and 𝑚∠𝑃𝐶𝑄 = 𝜃°.

A rotation of 0 degrees around the center 𝐶 is the identity transformation (i.e., for all points 𝐴 in the plane, it is the rotation defined by the equation 𝑅𝐶,0(𝐴) = 𝐴).

A rotation of 180° around the center 𝐶 is the composition of two rotations of 90° around the center 𝐶. It is also the transformation that maps every point 𝑃 (other than 𝐶) to the other endpoint of the diameter of a circle with center 𝐶 and radius 𝐶𝑃.

• A rotation leaves the center point 𝐶 fixed. 𝑅𝐶,𝜃 (𝐶) = 𝐶 states exactly that. The rotation function 𝑅 with center point 𝐶 that moves everything else in the plane 𝜃°, leaves only the center point itself unmoved.
• Any other point 𝑃 in the plane moves the exact same degree arc along the circle defined by the center of rotation and the angle 𝜃°.
• Then 𝑅𝐶,𝜃 (𝑃) is the point 𝑄 that lies in the counterclockwise half-plane of ray ⃗𝐶𝑃⃗⃗⃗⃗ such that 𝐶𝑄 = 𝐶𝑃 and such that 𝑚∠𝑃𝐶𝑄 = 𝜃°. Visually, you can imagine rotating the point 𝑃 in a counterclockwise arc around a circle with center 𝐶 and radius 𝐶𝑃 to find the point 𝑄.
• All positive angle measures 𝜃 assume a counterclockwise motion; if citing a clockwise rotation, the answer should be labeled with CW. A composition of two rotations applied to a point is the image obtained by applying the second rotation to the image of the first rotation of the point. In mathematical notation, the image of a point 𝐴 after a composition of two rotations of 90° around the center 𝐶 can be described by the point 𝑅 𝐶,90 (𝑅 𝐶,90 (𝐴)). The notation reads, “Apply 𝑅 𝐶,90 to the point 𝑅 𝐶,90 (𝐴).” So, we lose nothing by defining 𝑅 𝐶,180 (𝐴) to be that image. Then, 𝑅 𝐶,180 (𝐴) = 𝑅 𝐶,90 (𝑅 𝐶,90 (𝐴)) for all points 𝐴 in the plane.

In fact, we can generalize this idea to define a rotation by any positive degree: For 𝜃° > 180°, a rotation of 𝜃° around the center 𝐶 is any composition of three or more rotations, such that each rotation is less than or equal to a 90° rotation and whose angle measures sum to 𝜃°. For example, a rotation of 240° is equal to the composition of three rotations by 80° about the same center, the composition of five rotations by 50°, 50°, 50°, 50°, and 40° about the same center, or the composition of 240 rotations by 1° about the same center.

Notice that we have been assuming that all rotations rotate in the counterclockwise direction. However, the inverse rotation (the rotation that undoes a given rotation) can be thought of as rotating in the clockwise direction. For example, rotate a point 𝐴 by 30° around another point 𝐶 to get the image 𝑅 𝐶,30 (𝐴). We can undo that rotation by rotating by 30° in the clockwise direction around the same center 𝐶. Fortunately, we have an easy way to describe a rotation in the clockwise direction. If all positive degree rotations are in the counterclockwise direction, then we can define a negative degree rotation as a rotation in the clockwise direction (using the clockwise half-plane instead of the counterclockwise half-plane). Thus, 𝑅 𝐶,-30 is a 30° rotation in the clockwise direction around the center 𝐶. Since a composition of two rotations around the same center is just the sum of the degrees of each rotation, we see that

𝑅 𝐶,-30 (𝑅 𝐶,30 (𝐴)) = 𝑅 𝐶,0 (𝐴) = 𝐴,

for all points 𝐴 in the plane. Thus, we have defined how to perform a rotation for any number of degrees—positive or negative.

As this is our first foray into close work with rigid motions, we emphasize an important fact about rotations. Rotations are one kind of rigid motion or transformation of the plane (a function that assigns to each point 𝑃 of the plane a unique point 𝐹(𝑃)) that preserves lengths of segments and measures of angles. Recall that Grade 8 investigations involved manipulatives that modeled rigid motions (e.g., transparencies) because you could actually see that a figure was not altered, as far as length or angle was concerned. It is important to hold onto this idea while studying all of the rigid motions.

Constructing rotations precisely can be challenging. Fortunately, computer software is readily available to help you create transformations easily. Geometry software (such as Geogebra) allows you to create plane figures and rotate them a given number of degrees around a specified center of rotation. The figures in the exercises were rotated using Geogebra. Determine the angle and direction of rotation that carries each pre-image onto its (dashed-line) image. Assume both angles of rotation are positive. The center of rotation for Exercise 1 is point 𝐷 and for Figure 2 is point 𝐸.

Exercises 1–3

• To determine the angle of rotation, you measure the angle formed by connecting corresponding vertices to the center point of rotation. In Exercise 1, measure ∠𝐴𝐷′𝐴′. What happened to ∠𝐷? Can you see that 𝐷 is the center of rotation, therefore, mapping 𝐷′ onto itself? Before leaving Exercise 1, try drawing ∠𝐵𝐷′𝐵′. Do you get the same angle measure? What about ∠𝐶𝐷′𝐶′? Try finding the angle and direction of rotation for Exercise 2 on your own.
• Did you draw ∠𝐷𝐸𝐷′ or ∠𝐶𝐸𝐶′? Now that you can find the angle of rotation, let’s move on to finding the center of rotation. Follow the directions below to locate the center of rotation, taking the figure at the top right to its image at the bottom left
• a. Draw a segment connecting points 𝐴 and 𝐴′. b. Using a compass and straightedge, find the perpendicular bisector of this segment. c. Draw a segment connecting points 𝐵 and 𝐵′. d. Find the perpendicular bisector of this segment. e. The point of intersection of the two perpendicular bisectors is the center of rotation. Label this point 𝑃. Justify your construction by measuring ∠𝐴𝑃𝐴′ and ∠𝐵𝑃𝐵′. Did you obtain the same measure?

Exercises 4–5 Find the centers of rotation and angles of rotation for Exercises 4 and 5.

Lesson Summary

• A rotation carries segments onto segments of equal length.
• A rotation carries angles onto angles of equal measure.

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Common Core: High School - Geometry : Transformation and Congruence of Rigid Motions: CCSS.Math.Content.HSG-CO.B.6

Study concepts, example questions & explanations for common core: high school - geometry, all common core: high school - geometry resources, example questions, example question #1 : transformation and congruence of rigid motions: ccss.math.content.hsg co.b.6.

Determine whether the statement is true or false.

For a translation to be considered rigid, the starting and ending figures must be congruent.

Recall that a rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types. Therefore, for the translation to be considered "rigid" the two figures must be congruent by definition of a rigid motion.

Therefore, the statement, "For a translation to be considered rigid, the starting and ending figures must be congruent." is true.

Example Question #2 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

Which of the following is NOT a rigid motion?

• Translation
• Glide Reflection

Recall that a rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types.

These basic type of rigid motions include the following:

Therefore, of the answer selections, "Expansion" is the term that is NOT a rigid motion.

None of the other answers

A rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types.

Regardless of which type of rigid motion occurred, the following is true about the triangles' angles:

The following is also true about the side lengths:

Example Question #51 : Congruence

Determine whether the statement is true or false:

A glide reflection is a synonym for a rigid motion translation.

A glide reflection is a rigid motion that occurs when a figure is translated a certain distance and then reflected or reflected and then translated. 

A translation is a rigid motion describing when a object is moved a certain distance.

A glide reflection contains a translation but it is not a synonym for translation therefore, the statement is false.

Example Question #6 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

All glide reflections are reflections.

A glide reflection is a rigid motion that occurs when a figure is translated a certain distance and then reflected or reflected and then translated. In either case, a glide reflection aways contains a reflection.

Therefore, the statement, "All glide reflections are reflections." is true.

Sally has a quarter that is face up on the desk. If she slides it to Bob on the other side of the desk and he flips it over so that the tails side is facing up, is it a rigid motion?

In the situation where Sally has a quarter that is face up on the desk. The coin is the object. Then she slides it to Bob on the other side of the desk and he flips it over so that the tails side is facing up. Therefore, since the coin maintains it shape it is undergoing a translation to reach the other side of the desk and a reflection to flip the coin. This describes a glide reflection which is in fact, a rigid motion.

Therefore, the statement is describing a rigid motion. The answer is "Yes"

Example Question #3 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

Sally has a quarter that is face up on the desk. Then she slides the coin to Bob on the other side of the desk and he flips it over so that the tails side is facing up. What type of rigid motion does this situation describe?

This is not a rigid motion.

Example Question #4 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

Jane and Bob are filling up water balloons for a party they are throwing. Jane thinks the balloons should have more water in them so she fills them fuller. Each water balloon's circumference is one inch greater than before. Does this describe a rigid motion?

Recall that a rigid motion is that that preserves the distances between points within the object while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types. The water balloons are the objects in this situation. Since the they are filled with more water to increase their circumferences, it does not preserve the shape of the object and thus, is not a rigid motion. 

Select the answer that best completes the following definition:

A motion that preserves distance in the plane is called a __________ . 

Non-Rigid Motion

Rigid Motion

Transformation

Therefore, "A motion that preserves distance in the plane is called a rigid motion ".

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Common Core Geometry Practice & Activity (Rotations G.CO.3, G.CO.4, G.CO.5)

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Transformation Worksheets: Reflection, Translation & Rotation

Geometry transformation worksheet: reflection, translation, rotation – free & printable.

Help your students understand the art of decoding the look of an image when rotated or reflected or flipped with our free & Printable worksheets

Grade 3 transformation sample questions.

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Big Ideas Math Geometry Answers | Big Ideas Math Book Geometry Answer Key

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