Definitions of Transformations
Student Activity
Name____________________
Class____________________
Overview
A transformation is a one to one function (also called a mapping) that maps points in the plane to points in
the plane. A transformation is called an isometry or rigid motion, if the distance between points is
preserved (unchanged following the transformation). The following lesson is an investigation of the
defining properties of three important basic transformations: translations, rotations, and reflections. Our
objectives are to discover the definition of each of these transformations and to show that each is an
isometry and therefore will map a triangle onto a congruent triangle.
Materials
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TI-Nspire™ handheld
Definitions_of_Transformations_Student.tns
ACTIVITY 1 – Translations
Problem 1.1 – Definition of a Translation
Open Definitions_of_Transformations_Student.tns, read page 1.2 and
then turn to page 1.3. This page illustrates ABC .
1. Use the Points & Lines Menu to construct a vector v .
2. Use the Transformation Menu to find the image of
ABC
under a translation through vector v and label the image
A ' B ' C ' . (Notation: Tv (ABC )  A ' B ' C ' .)
3. Draw lines AA ' , BB ' and CC ' . What appears to be true?
4. “Grab and drag” the point at the end of vector, v , to change the translation.
 Make a conjecture.
5. Measure the length of vector v . Record your measurements in the following table.
6. Measure the distance from point A to point A ' ; B to B ' ; C to C ' . Record your measurements
in the following table.
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Name____________________
Definitions of Transformations
Student Activity
Class____________________
7. “Grab and drag” the point at the end of vector, v , to change the position of the vector. Record your
measurements in the following table.
Position of Vector v
v
AA '
BB '
CC '
PP '
Position 1
Position 2
Position 3

What do you observe?
8. Construct a point P anywhere in the plane and find its image P ' under Tv .
 What do you notice about the distance PP ' ? What if you change vector, v ?
9. Using your own words write a definition of Tv .
.
Problem 1.2 – Properties of Translations
1. Measure line segments AB and A ' B ' . What appears to be true? Repeat for line segments AC
and A ' C ' , and for line segments BC and B ' C ' .
2. “Grab and drag” the point at the end of vector, v , to change the translation. What do you observe?
Make a conjecture.
3. Measure angles ABC and A ' B ' C ' . What appears to be true? Repeat for angles ACB and
A ' C ' B ' , and for angles CAB and C ' A ' B ' .
4. “Grab and drag” the point at the end of vector, v , to change the translation. What do you observe?
Make a conjecture.
5. Make a conjecture about a triangle and its image under a translation.
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Name____________________
Definitions of Transformations
Student Activity
Class____________________
ACTIVITY 2 – Rotations
Problem 2.1 – Definition of Rotation
Read page 2.1 and turn to page 2.2. This page illustrates ABC and an angle,  .
1. Use the Points & Lines Menu to construct a point P not on ABC .
2. Use the Transformation Menu and find the image of
angle,  . (Notation: RP , (ABC )  A ' B ' C ' .)
ABC under a Rotation about P through
3. Draw line segments PA and PA ' . What appears to be true? Confirm your conjecture.
4. “Grab and drag” the open point on the angle and move it to several different positions to change
the value of  . What do you observe?
5. Repeat for line segments, PB and PB ' and then again for
PC and PC ' . Make a conjecture.
6. Measure APA ' , BPB ' and CPC ' . Record your measurements in the table below.
7. “Grab and drag” the open point on the angle and move it to several different positions to change
the value of  . Record your measurements in the table below.
Position of θ

m APA '
m BPB '
m CPC '
Position 1
Position 2
Position 3

Make a conjecture.
8. Construct a point Q anywhere in the plane and find its image, Q ' under RP , .
Note: Because you are rotating a point around a point, the order of choice of points must be,
first the center of the rotation,
P and then the preimage point ( Q ) and then the value of  .

What do you notice about the distances QP and Q ' P ?

What do you notice about the measure of QPQ ' ?
9. Find the image of point P under RP , . Make an observation.
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Definitions of Transformations
Student Activity
Name____________________
Class____________________
10. Using your own words write a definition of RP , .
Problem 2.2 – Properties of Rotations
1. Measure line segments AB and A ' B ' . What appears to be true? Repeat for line segments AC
and A ' C ' , and for line segments BC and B ' C ' .
2. “Grab and drag” the open point on the angle to change the value of θ. Make a conjecture.
3. Measure angles ABC and A ' B ' C ' . What appears to be true? Repeat for angles ACB and
A ' C ' B ' , and for angles CAB and C ' A ' B ' .
4. “Grab and drag” the open point on the angle to change the value of  . Make a conjecture.
5. Make a conjecture about a triangle and its image under a rotation.
Activity 3 – Reflections
Problem 3.1 – Definition of a Reflection
Read page 3.1 and turn to page 3.2. This page illustrates ABC .
1. Use the Points & Lines Menu to construct a line, l .
2. Use the Transformation Menu and find the image of
ABC
under a reflection over line, l .
(Notation: Rl (ABC )  A ' B ' C ' .)
3. Draw lines AA ' , BB ' and CC ' . What appears to be true?
4. “Grab and drag” the line, l , to change the reflection.
 Make a conjecture.

How could you test your conjecture?
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Definitions of Transformations
Student Activity
5. Construct a point

Class____________________
P anywhere on line, l , and find its image P ' under Rl
What do you observe?
6. Construct a point

Name____________________
P anywhere in the plane and find its image P ' under Rl
What do you observe?
7. Using your own words write a definition of
Rl
Problem 3.2 – Properties of Reflections
1. Measure line segments AB and A ' B ' .
 What appears to be true?
Repeat for line segments AC and A ' C ' , and for line segments BC and B ' C ' .
2. “Grab and drag” the line, l, to change the reflection.
 Make a conjecture.
3. Measure angles ABC and A ' B ' C ' . What appears to be true? Repeat for angles ACB and
A ' C ' B ' , and for angles CAB and C ' A ' B ' .
4. “Grab and drag” the line, l, to change the reflection.
 Make a conjecture.
5. Make a conjecture about a triangle and its image under a reflection.
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