 Instructional Unit: Transformation

A transformation is a way of moving or changing a figure. There are three types of basic transformations that preserves the size and shape of the figure. These three, reflections, rotations, and translations, are called isometrics. Isometrics flip, slide, or turn a figure but never bend or distort it. In this activity, you'll experiment with basic isometrics by transforming a flag -shaped polygon.

Investigate: Translations

1. Construct the vertices of a flag shape and construct its interior.

2. In order to translate a shape, you need to indicate a direction and a distance. To do this, construct segmement FG. Then select, in order, point F and point G. In the transform menu, choose Mark Vector. 3. Select the interior of the flag; then. In the Transform menu choose Translate. Make sure By Marked Vector is checked in the Translate dialog box, theb click OK.

4. Change the color or shading of the translate image

5. Display the translated polygon's labe1, then change the label to Translate Image

6. Drag pont G to change your vector and observe the relationship between the translated image and the original figure.

Q1. Compare the translated image to the original figure. How are they different and how are they the same

Sketch and Investigation: Rotation.

7. In order to rotate a shape, you need to indicate a center of rotation and an angle of rotation. Start by creating angle JHI using two attached segments, as indicated at the figure. 8. Mark angleJHI as an angle of rotation

9. Mark angle JHI as an angle of rotation.

10. Rotate the original flagshaped interior by the marked angle

11. Change the color or shading of the rotated image. Also
display its label and change the label to Rotated image. Also display its label and change the label to Rotated image.

12. Drag point I to change your angle, and observe the relationship between the rotated image and the original figure.

Q2. Compare the rotated image to the original figure. How are they different and how are they the same?

Sketch and Investigate: Reflections
13. To reflect a shape, you need a mirror line (also called a line of reflection). Draw a line and label it Mirror line.

14. Mark the line as a mirror.

15. Reflect the original flag-shaped interior. Your image may end up off the screen. If it does, move the original figure closer to the mirror line.

16. Change the color or shading of the reflected image. Label it Reflected image.
17. Drag your mirror line, and observe the relationship between the reflected image and the original figure. Q3. Compare the reflected image to the original figure. How are they different and how are they the same?
Q4. Explain whether it is possible for any of the three images in your sketch to lie directly on top of one another. Experiment by dragging different parts of your sketch.

Explore More
1. Use reflections, rotations, translations, or combinations of these transformations to make a design.
2. Reflect a figure over a line, then reflect the image over a second line that intersects the first. What single transformation would take your original figure to the second reflected image?
3. Reflect a figure over a line, then reflect the image over a second line that is parallel to the first. What single transformation is the same as this combination of two reflections?

Properties of Reflection

When you look at yourself in a mirror, how far away does your image in the mirror appear to be? Why is it that your reflection looks just like you, but backwards? Reflections in geometry have some of the same properties of reflections you observe in a mirror. In this activity, you'll investigate the properties of reflections that make a reflection the "mirror image" of the original.
Sketch and Investigate: Mirror Writing
1. Construct vertical line AB
2. Construct point C to the right of the line.
3. Mark AB as a mirror. 4. Reflect point C to construct point C'.
5. Turn on Trace Points for points C and C'.
6. Drag point C so that it traces out your name.
Q1 What does point C' trace?

7. For a real challenge, try dragging point C' so that point C traces out your name.

Sketch and Investigate: Reflecting Geometric Figures

8. Turn off Trace Points for points C and C'.
9. Construct triangle CDE.
Diagram
10. Reflect triangle CDE (sides and vertices) over AB.
11. Drag different parts of either triangle and observe how the triangles are related. Also drag the mirror line. 12. Measure the lengths of the sides of triangles CDE and C'D'E'.
13. Measure one angle in triangle CD E and measure the corresponding angle in triangle C'O'E'.

Q2 What effect does reflection have on lengths and angle measures?

Q3 Are a figure and its mirror image always congruent? State your answer as a conjecture.

Q4 Going alphabetically from C to D to E in triangle CDE, are the vertices oriented in a clockwise or counterclockwise direction? In what direction (clockwise or counterclockwise) are vertices C',D', and E'
oriented in the reflected triangle?

14. Construct segments connecting each point and its image: C to C', D to D', and E to E'. Make these segments dashed. 15. Drag different parts of the sketch around and observe relationships between the dashed segments and the mirror line.

Q5 How is the mirror line related to a segment connecting a
point and its reflected image?

Explore More

1. Suppose Sketchpad didn't have a Transform menu. How could you construct a given point's mirror image over a given line? Try it. Start with a point and a line. Come up with a construction for the reflection of the point over the line using just the tools and the Construct menu. Describe your method.

2. Use a reflection to construct an isosceles triangle. Explain what you did.

Reflections in the Coordinate Plane

In this activity, you'll investigate what happens to the coordinates of points when you reflect them across the x- and y-axes in the coordinate plane.
1. Show the grid.
2. Draw triangle CDE with vertices on the grid
3. Measure the coordinates of each vertex 4. Mark the y -xis as a mirror.
5. Reflect the triangle.
6. Measure the coordinates of the image's vertices.
7. Drag vertices to different points on the grid and look for a relationship between a point's coordinates and the coordinates of the reflected image across the y-axis.

Q1 Describe any relationship you observe between the coordinates of the vertices of your original triangle and the coordinates of their reflected images across the y-axis.

8. Now mark the x-axis as a mirror and reflect your original triangle. 9. Before you measure coordinates, can you guess
what they'll be? Measure to confirm.

Q2 Describe any relationship you observe C between the coordinates of the original points and the coordinates of their reflected images across the x- axis

Explore More

1. Draw a line on the grid that passes through the origin and makes a 45degree angle with the x-axis (in other
words, the line y = x). Reflect your triangle across this line. What do you notice about the coordinates of the vertices of this image?

Translations in the Coordinate Plane

In this activity, you'll investigate what happens to the coordinates of points when they're translated in the coordinate plane.
1. Show the grid.
2. Draw a segment from the origin to anywhere on the grid. Label the end point C

point C.
3. Measure the coor- dinates of point C.
4. Mark vector AC
5. Draw triangle triangle DEF with vertices on the grid 6. Translate the triangle by the marked vector.
7. Measure the coordinates of the two triangles' six vertices.
8. Experiment by dragging point C or any of the triangle vertices. Look
for a relationship between a point's coordinates and the coordinates of its image under a translation.

Q1. Where can you drag point C so that the original points and the corresponding image points always have the same y-coordinates but have different x-coordinates?

Q2. Where can you drag point C so that the original points and the corresponding image points always have the same x-coordinates but have different y-coordinates?

Q3. When the vector defined by the origin and point C translates your original triangle to the left and up, what must be true of the coordinates of point C?

Q4 Suppose point C has coordinates (a, b). What are the coordinates of the image of a point (x, y) under a translation by (a, b)?

Reflections over Two Parallel Lines

In this investigation, you'll see what happens when you reflect a figure
over a line then reflect the image over a second line parallel to the first,

Sketch and Investigate

1, Construct any irregular polygon interior.
2. Show the label of one of the polygon's vertices and change the label to A.

3. Construct a line EF. 4. Mark the line as a mirror and reflect the polygon and point A over it.

5. Construct point G and a line through point G parallel to EF
6. Mark this second line as a mirror and reflect the first reflected image and point A' over it.

7.Drag the original figure and the two lines and observe their relationships to the two images. Q1 Two reflections move your original figure to its second image. What single transformation do you think will do the same thing? (If you're not sure, go on to the next steps, then come back to this question.)

Reflections over Two Parallel Lines (continued)

8. Construct AA ".
9. Construct points H and I where AA" intersects the two lines.
10. Measure AA " and HI. Q2 Drag one of the lines and compare the two E distances. How are they related?
Q3 HI is the distance between the two lines. Why?
11. Mark AA " as a vector, then translate the original figure by this
vector.
Q4 Describe the result of Step 11, above. What single transformation is equivalent to the combination of two reflections over parallel lines?
Q5 Answer the following questions to explain why AA " and HI are related as they are:

a. How does AH compare to HA '?
b. How does A'I compare to IA "?

c. AA' + A' A" =
d. Complete the rest of the explanation on your own.
Explore More
1. In the same sketch, try reflecting your figure and then its image over the two lines in the opposite order. Describe the result.

Reflections over Two Intersecting Lines

In this investigation, you'll see what happens when you reflect a figure over a line and then reflect the image over a second line that intersects the first.

Sketch and Investigate

1. Construct any irregular polygon interior.
2. Show the label of one of the
polygon's vertices and change it to A.
3. Construct two intersecting lines and their point of intersection. 4. Mark the line closest to the polygon as a mirror, then reflect the polygon and the labeled point over this
line. Change the shading or color of the image. (See the figure below left. If necessary, move the polygon so
Steps 1-3 that the image falls between the lines.)
5. Mark the other line as a mirror, then reflect the image from the first j reflection over this second line. Change the shading or color of this second image. (See the figure below right.)  6. Drag the original figure and the two lines and observe their relationships to the two images.
Q1 Two reflections move your original figure to its second image. What single transformation do you think
will do the same thing? (If you're not sure, go on to the next steps, then come back to this question.)
7. Construct AI, where point A is a point on the original figure and point I is the point of intersection of the lines. 8. Construct AI' and IA"
9. Measure angle AIA"
10. Measure angle FIH, the angle between the lines.
Q2 Compare the two angle measures. How are they related?
11. Mark point I as a center for rotation and mark LAIA ", then rotate the original figure by this angle.
Q3 Describe the result of step 11. What single transformation is equivalent to the combination of two reflections
over intersecting lines? .
Q4 Answer the following questions to explain why m angle AIA" and
M angle FIH are related the way they are:
a. How does m angle AIF compare to m angle AIA'?
(Try to answer without measuring first.)
b. How does m angle A'IH compare to m angle A'IA"?
c. m angle AIA' + m angle A'IA" = m angle
d. Complete the rest of the explanation on your own. Use a separate sheet, if necessary,

Glide Reflections

In this activity you will investigate an isometry called a glide reflection. Glide reflection is not a transformation 'found in the Transform menu, I but you'll define it as a custom transformation, and in the process you'll learn what a glide reflection is and what it does.

Sketch and Investigate

1. Construct an irregular polygon interior, like polygon ABCD, shown at right.
2. Construct line EF.
3. Mark EF as a mirror and reflect the polygon interior across EF.
4. Construct a point G on the line so that E and G are about an '
inch apart. 5. Mark EG as a vector and translate the reflected image by the marked vector. This second image is a glide reflection of your original figure.
6. Drag point G to see how it affects the glide-reflected image.
7. Hide the intermediate image (the first reflection). 8. Select the glide-reflected image and the original image; then, in the Transform menu, choose Define Transform. Name the 2-step transformation "Glide Reflection."
9. Glide Reflection should now be a command in the Transform menu. Use Glide Reflection to construct several more images, as shown at right.
10. Drag parts of your sketch (vertices of the original polygon, point G, points E and F, the line) and observe the effects.

Glide Reflections (continued)

Q1 A glide reflection is the product of two transformations. What are they?

Q2 A translation is also the product of two transformations. What are they?

Q3 A glide reflection can be thought of as a product of what three transformations?

Explore More

1. Create a polygon that looks like a foot, then use glide reflections to make a sketch that looks like footprints in sand. To make the footprints appear sequentially, as if made by a walking invisible person, follow these steps:

(a) Make Hide/Show action buttons for each footprint.
b. Double-click on each Show button with the Information tool (the question mark) and give each a 1 second
in-sequence delay.
c. Make a Sequence button with the Show buttons.
d. Sequence the Hide buttons to hide all the footprints.
2. Experiment with reflections across three random lines. Does this
produce a glide reflection?
3. What's the product of a rotation and a reflection? What's the product of a rotation and a translation?

Symmetry in Regular Polygons

A figure has reflection symmetry if you can reflect the figure over a line so that the image will coincide with the original figure. The line you reflect over is called a line of symmetry or a mirror line. A figure has rotational symmetry if you can rotate it some number of degrees about some point so that the rotated image will coincide with the original figure. In this exploration, you'll look for reflection and rotation symmetries of regular polygons.

Sketch and Investigate

1. Construct a regular polygon and its interior. You can use an equilateral triangle, square, a regular pentagon, or a regular hexagon. You may want to have different groups in your class investigate different shapes. 2. Construct a line.

3. Mark the line as a mirror and reflect the polygon interior over it.
4. Give the image a lighter shade.

5. Drag the line until the image of your polygon coincides exactly with the original.
Q1 When a reflection image coincides with the original figure, the reflection line is a line of symmetry. Describe how the line of symmetry is positioned relative to the figure.
6. Drag the line so that it is a different line of symmetry. Repeat until you have found all the reflection symmetries of your polygon.
Q2 Fill in one entry in the table below: the number of reflection symmetries for your polygon. (Note: Be careful not to count the same line twice!) You'll come back to fill in other entries as you gather more information.

 Number of sides of regular polygon 3 4 5 6 7 .. n Number of reflection symmetry .. Number of rotation symmetry ..

Symmetry in Regular Polygons (continued)

Next, you'll look for rotation symmetries.
7. Move the line so that the reflected Image is out of the way.
8. If the polygon's center doesn't already
exist, construct it. 9. Use the Segment tool to construct an angle.
10. Measure the angle.
11. In Preferences, set the Angle Unit to directed degrees.
12. Mark the center of the polygon as a center for rotation and mark
the angle measurement. Rotate the polygon interior by this marked angle measurement.
13. Give the rotated image a different shade.
14. Change the angle so that the rotated image fits exactly over the original figure.
Q3 What angle measure causes the figures to coincide? Polygon:......... Rotation angle: .........
15. Continue changing your angle to find all possible rotation symmetries of your polygon.
Q4 Count the number of times the rotated image coincides with the original when rotating from 0° to 180° and from -180° back to 0°. In your chart on the preceding page, record the total number of rotation symmetries you found. Note: Count one complete revolution as one of your rotation symmetries.
Q5 Combine the results from other members of your class to complete your chart with the reflection and rotation symmetries of other regular polygons.
Q6 Use your findings to write a conjecture about the reflection and rotation symmetries of a regular n-gon. Include in your conjecture a statement about the smallest angle of rotational symmetry greater
than zero.