Defining Rigid Transformations
The Lesson Activities will help you meet these educational goals:
· Content Knowledge—You will develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
· Mathematical Practices—You will reason abstractly and quantitatively and use appropriate tools strategically.
· Inquiry—You will perform an investigation in which you will make observations and draw conclusions.
· STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations.
· 21st Century Skills—You will assess and validate information.
Directions
You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.
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Self-Checked Activities
Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.
1. Rotations
Use GeoGebra to explore the properties of rotations and complete each step below. If you need help, follow these instructions for using GeoGebra.
a. Move the slider to rotate At which angles does coincide with and
Sample answer:
Triangle / Angle0°
45°
115°
210°
b. Create a circle with center P that passes through A. Create two more circles, one passing through B and the other through C. Use the slider again to rotate
As the triangle rotates, what do you observe with respect to the circles you made?
Sample answer:
As rotates, each of the vertices moves in a circular arc along one of the three circles.
c. Measure the angles of Then measure the angles of when coincides with the other three triangles. Record the measure of each angle in the table.
Sample answer:
∆ABC / Measure / ∆DEF / Measure / ∆GHI / Measure / ∆JKL / Measure108° / 108° / 108° / 108°
27° / 27° / 27° / 27°
45° / 45° / 45° / 45°
d. How do the measures of the angles ofcompare with those of when coincides with the other three triangles? What can you say about the preservation of the angle measurements of a shape during a rotation?
Sample answer:
The measure of each angle of is equal to the corresponding angle of when coincides with the other three triangles. The measures of the angles of the triangle are preserved as the figure rotates.
e. Measure the lengths of the sides of. Then measure the lengths of the sides of when coincides with the other three triangles. Record the measurements in the table.
Sample answer:
∆ABC / Length / ∆DEF / Length / ∆GHI / Length / ∆JKL / Length1.41 / 1.41 / 1.41 / 1.41
2.24 / 2.24 / 2.24 / 2.24
3 / 3 / 3 / 3
f. How do the measures of the side lengths ofcompare with those of whencoincides with the other three triangles? What can you say about the preservation of the side lengths of a shape during a rotation?
Sample answer:
The measure of each side of is equal to the corresponding side of whencoincides with the other three triangles. The side lengths of the triangle are preserved as the figure rotates.
2. Reflections
Use GeoGebra to explore the properties of reflections and complete each step below.
a. Polygon ABCD has reflected about to give polygon A′B′C′D′. Change the line of reflection. You can move each endpoint individually or move the entire line. As you move the line of reflection, what do you observe about the relationship between polygon ABCD and its image?
Sample answer:
As moves, the reflected image changes its position with respect to the position of the line. The polygons will touch one another if the line of reflection touches polygon ABCD.
b. Be sure that polygon ABCD is not touching its image. Draw a line segment connecting point A and point AGH. Notice that intersects the line of reflection, Mark the point of intersection of and Measure and record the angle at the intersection of and What happens if you move the line of reflection?
Sample answer:
The measure of the angle of intersection for and is 90°. As the line of reflection changes position, the angle measure stays the same.
c. Measure and record the length of Then measure and record the distance between point A and the point of intersection you created. Do the same between point AGH and the same intersection. With the line segments in place, move the line of reflection, What do you notice about the measurements? Complete the table and describe what you see using geometric terms.
Sample answer:
The distance from point A to the line of reflection is the same as the distance from point AGH to the line of reflection. This means that the line of reflection bisects
A to AGH / A to point of intersection / AGH to point of intersection4 / 2 / 2
5.77 / 2.89 / 2.89
7.9 / 3.95 / 3.95
d. Repeat parts b and c for corresponding points B and BGH, C and CGH, and D and DGH. With your measurements displayed in GeoGebra, move the line of reflection, around the coordinate plane. Also, move the points of the preimage ABCD as you’d like. Based on your investigation, what can you conclude about the line of reflection with respect to the image and the preimage? Express your answer in geometric terms.
Sample answer:
A vertex on polygon ABCD and its corresponding vertex on polygon AGHBGHCGHDGH are always equidistant from the line of reflection. Each of the four line segments ( for example) is perpendicular to and cut in half by the line of reflection, This means that is a perpendicular bisector of those line segments.
e. It’s time to verify that a reflection is a rigid transformation. You’ll begin with the angle measurements. Measure the angles of polygon ABCD and the angles of the reflected image. Record the measure of each angle to the nearest degree.
Sample answer:
ABCD / Measure / AGHBGHCGHDGH / Measure90° / AGHBGHCGH / 90°
45° / BGHCGHDGH / 45°
135° / CGHDGHAGH / 135°
90° / DGHAGHBGH / 90°
f. Now measure the lengths of the sides of polygon ABCD and the lengths of the sides of the reflected image. Record the measurements in the table.
Sample answer:
ABCD / Length / AGHBGHCGHDGH / Length1.41 / 1.41
2.89 / 2.89
2 / 2
1.41 / 1.41
g. How do the angle measures of polygon ABCD compare with the angle measures in the reflected image? How do the side lengths compare? What can you say about the preservation of the angle measurements and side lengths of a shape during a reflection?
Sample answer:
The measures of the interior angles of polygon ABCD are equal to the measures of the corresponding angles in the reflected image. This means that angle measurements of the polygon are preserved as the image is reflected. The same is true for the side lengths. The measures of both angles and sides are preserved in a reflection. A reflection is a rigid transformation.
3. Translations
Use GeoGebra to explore the properties of translations and complete each step below.
a. Begin by translating polygon ABCD. Create a line segment of length 5 units to the right of points A, B, C, and D. Connect the points you created to form a polygon that is congruent to ABCD. When you’re done, you’ll have translated polygon ABCD to the right by 5 units.
Measure the slopes of the line segments (i.e., construction paths) you created. Record the measurements in the table.
Sample answer:
The vertices may be different, but the slope of each line is m = 0.
Line Segment / Slopem = 0
m = 0
m = 0
m = 0
b. How do the slopes of the line segments in part a compare with one another? In geometric terms, what can you conclude about the lines?
Sample answer:
All the lines have the same slope, m = 0, showing that they are all parallel.
c. Add another set of points that are 8 units below points A, B, C, and D. This time, you’ll need to rotate the lines into a different orientation. Connect the points you created to form another polygon that is congruent to ABCD. When you’re done, you’ll have translated polygon ABCD down by 8 units.
Measure the slopes of the line segments (i.e., construction paths) you created. Record the measurements in the table.
Sample answer:
The vertices may be different, but the slope of each line is undefined.
Line Segment / Slopeundefined
undefined
undefined
undefined
d. How do the slopes of the line segments in part c compare with one another? Based on your observations in parts a through c, what can you say about the relationship between a preimage and an image under a translation?
Sample answer:
All of the lines have an infinite slope because they are all vertical. They are all parallel lines. Under a translation, the corresponding points in an image and the preimage are connected by parallel paths of equal length.
e. It’s time to verify that a translation is a rigid transformation. Begin with the angle measurements. Measure the angles of polygon ABCD and the angles of the reflected image. Record the measure of each angle to the nearest degree.
Sample answer:
This answer assumes that the newly formed polygons are EFGH and IJKL.
ABCD / Measure / EFGH / Measure / IJKL / Measure90° / 90° / 90°
63° / 63° / 63°
153° / 153° / 153°
53° / 53° / 53°
f. How do the measures of the angles of polygon ABCD compare with those of the translated images? Based on your observation, what can you conclude about preservation of angle measurements during a translation?
Sample answer:
The measures of the angles of polygon ABCD are equal to the measures of the corresponding angles of both translated images. This means that the angles of the polygon are preserved as the figure is translated in any direction.
g. Now measure the lengths of the sides of polygon ABCD and the lengths of the sides of the translated images. Record the measurements in the table.
Sample answer:
This answer assumes that the newly formed polygons are EFGH and IJKL.
ABCD / Length / EFGH / Length / IJKL / Length3.16 / 3.16 / 3.16
3.16 / 3.16 / 3.16
1.41 / 1.41 / 1.41
3.16 / 3.16 / 3.16
h. How do the side lengths of polygon ABCD compare with those of the translated images? What can you say about preservation of the measurements of side lengths when a figure is translated?
Sample answer:
The side lengths of polygon ABCD are equal to corresponding side lengths in the translated images. This means that lengths of sides of the polygon are preserved as the figure is translated in any direction.
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