unit 3 transformations/rigid...
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Unit 3 β Transformations/Rigid Motions
Day Classwork Homework
Wednesday
9/27 Unit 2 Test
Thursday
9/28
Reflections, Rotations, Translations
in the Coordinate Plane
HW 3.1
Friday
9/29
Reflections, Rotations, Translations
in the Coordinate Plane
HW 3.1 (continued)
Monday
10/2
Reflections HW 3.2
Tuesday
10/3
Rotations
HW 3.3
Wednesday
10/4
Rotations
Unit 3 Quiz 1
HW 3.4
Thursday
10/5
Symmetry HW 3.5
Friday
10/6
Translations
Unit 3 Quiz 2
HW 3.6
Monday
10/9 No School
Tuesday
10/10
Translations
HW 3.7
Wednesday
10/11
Review
Unit 3 Quiz 3
Review Sheet
Thursday
10/12
Review Review Sheet
Friday
10/13 Unit 3 Test
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CONGRUENCE TRANSFORMATIONS
A transformation πΉof the plane is a function that assigns to each point π of the plane a unique point πΉ(π) in the
plane. Transformations that preserve lengths of segments and measures of angles are called basic rigid
motions. We call a figure that is about to undergo a transformation the pre-image while the figure that has
undergone the transformation is called the image.
Congruence Transformations Definition Example
Reflection
Translation
Rotation
Using the figures above, identify specific information needed to perform the rigid motion shown.
For a rotation, we need to know:
For a reflection, we need to know:
For a translation, we need to know:
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TRANSFORMATIONS IN THE COORDINATE PLANE
A reflection or βflipβ is a transformation in a line called the line of reflection. Each point of the pre-image
and its corresponding point on the image are the same distance from this line. Examples Triangle JKL has vertices J(0, 3), K(-2, -1), and L(-6, 1). Graph triangle JKL and its image in the given line.
a. x = -4 b. y = 2
Reflections Symbols Example
Reflection in the x-axis To reflect a point in the x-axis, multiply its y-coordinate by -1
Reflection in the y-axis To reflect a point in the y-axis, multiply its x-coordinate by -1.
Examples Graph each figure and its image under the given reflection.
a. βπ΄π΅πΆ with vertices A (-5, 3), B (2, 0), and C (1, 2) in the x-axis
b. βπ΄π΅πΆ in the y β axis.
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Reflections
Symbols Example Reflection in the line y = x
To reflect a point in the line y = x, interchange the x and y
coordinates
Reflection in the line y = -x To reflect a point in the line y = -x, interchange and change the sign
of the x and y coordinates.
Examples
a. Quadrilateral JKLM has vertices J (2, 2), K (4, 1), L (3, -3), and M (0, -4). Graph JKLM and its image JβKβLβMβ in the line y = x.
A rotation or βturnβ moves every point of a pre-image through a specified angle and direction about a fixed point. When a point is rotated a 90β°, 180β°, or 270β° counterclockwise about the origin, you can use the
following rules.
Rotations in the Coordinate Plane
Type of Rotation Symbols Examples
90β° Rotation
180β° Rotation
270β° Rotation
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Examples Triangle PQR has vertices P(1, 1), Q (4, 5), and R (5, 1). Graph triangle PQR and its image after a rotation 90β° about the origin.
Parallelogram FGHJ has vertices F (2, 1), G (7, 1), H (6, -3), and J (1, -3). Graph FGHJ and its image after a rotation 180β° about the origin.
Triangle JKL has vertices J(3, -7), K (1, -1), and L (5, -3). Graph triangle JKL and its image after a rotation 270β° counterclockwise about the origin.
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A slide is a transformation that moves all points of a figure the same distance in the same direction. Since vectors can be used to describe both distance and direction, vectors can be used to define translations.
Translation in the Coordinate Plane
Words Symbols Example To translate a point along vector <a, b>, add a to the x-coordinate
and b to the y-coordinate
Examples Graph each figure and its image along the given vector.
a. βπΈπΉπΊ with vertices E (-7, -1), F (-4, -4), and G (-3, -1); <2, 5>
b. Square JKLM with vertices J (3, 4), K (5, 2), L (7, 4), and M (5, 6); <-3, -4>
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REFLECTIONS
Construct the segment that represents the line of reflection
for quadrilateral π΄π΅πΆπ· and its image π΄β²π΅β²πΆβ²π·β².
What is true about each point on π΄π΅πΆπ· and its
corresponding point on π΄β²π΅β²πΆβ²π·β²?
What term is used to describe the line of reflection? For a line πΏ in the plane, a reflection across πΏ is the transformation ΞπΏ of the plane defined as follows:
1. For any point π on the line πΏ, ΞπΏ(π) = π, and
2. For any point π not on πΏ, ΞπΏ(π) is the point π so that πΏ is the perpendicular bisector of the
segment ππ.
Constructing the line of reflection
Construct the line of reflection across which each image below was reflected.
1. 2.
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Constructing an image under a given reflection
1. Construct circle πΆπ΄: center π΄, with radius such that the circle crosses π·πΈΜ Μ Μ Μ at two points
(Label the points πΉ and πΊ).
2. Construct circle πΆπΉ: center πΉ, radius πΉπ΄Μ Μ Μ Μ and circle πΆπΊ: center πΊ, radius πΊπ΄Μ Μ Μ Μ . Label the [unlabeled] point
of intersection between circles πΉ and πΊ as point π΄β². This is the reflection of vertex π΄ across π·πΈΜ Μ Μ Μ .
3. Repeat steps 1 and 2 for vertices π΅ and πΆ to locate π΅β² and πΆβ².
4. Connect π΄β², π΅β², and πΆβ² to construct the reflected triangle.
Practice 1. Reflect π΄π΅πΆπ· across πΈπΉΜ Μ Μ Μ .
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Reflect the given image across the line of reflection provided.
4.
5. Draw a triangle β³ π΄π΅πΆ. Draw a line π through vertex πΆ so that it intersects the triangle at more than just the vertex. Construct the reflection across π.
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ROTATIONS For 0Λ < π < 180Λ, the rotation of π degrees around the center πΆ is the transformation π πΆ,π of the plane
defined as follows: 1. For the center point πΆ, π πΆ,π(πΆ) = πΆ, and 2. For any other point π, π πΆ,π(π) is the point π that lies in the counterclockwise half-plane of ray
πΆπββββ β, such that πΆπ = πΆπ and β ππΆπ = πΛ. A rotation of 0Λ around the center πΆ is the identity transformation, i.e., for all points π΄ in the plane, it is the rotation defined by the equation π πΆ,0(π΄) = π΄. A rotation of 180Λ around the center πΆ is the composition of two rotations of 90Λ around the center πΆ. It is also the transformation that maps every point π (other than πΆ) to the other endpoint of the diameter of circle with center πΆ and radius πΆπ. Finding the angle and direction of rotation:
1. Find the measure of β π΄π·β²π΄β².
b. What happened to β π·?
c. Find the measures of β π΅π·β²π΅β² and β πΆπ·β²πΆβ². What do you notice?
2. Find the angle and direction of rotation. Describe your process.
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Finding the center of rotation.
1. Draw a segment connecting points π΄ and π΄β².
2. Using a compass and straightedge, find the perpendicular bisector of this segment.
3. Draw a segment connecting
points π΅ and π΅β².
4. Find the perpendicular bisector of this segment.
5. The point of intersection of the
two perpendicular bisectors is the center of rotation. Label this point π.
Justify your construction by measuring angles β π΄ππ΄β and β π΅ππ΅β. Did you obtain the same measure? Exercises: Find the center of rotation and angle of rotation for each pair of figures below.
1. 2.
Lesson Summary:
A rotation maps segments onto segments of equal length.
A rotation maps angles onto angles of equal measure.
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More Practice with Rotations
1. Rotate the triangle π΄π΅πΆ 60Β° around point πΉ using a compass and straightedge only.
2. Rotate quadrilateral π΄π΅πΆπ· 120Β° around point πΈ using a compass and straightedge.
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3. On your paper, construct a 45Β° angle using a compass and straightedge. Rotate the angle 180Β° around its vertex, again using only a compass and straightedge. What figure have you formed, and what are its angles called?
4. Draw a triangle with angles 90Β°, 60Β°, and 30Β° using only a compass and straightedge. Locate the midpoint of the longest side using your compass. Rotate the triangle 180Β° around the midpoint of the longest side. What figure have you formed?
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5. On your paper, construct an equilateral triangle. Locate the midpoint of one side using your compass. Rotate the triangle 180Β° around this midpoint. What figure have you formed?
6. Use the initials provided below. If you create your own WordArt initials, copy, paste, and rotate to create a design similar to the one below. Find the center of rotation and the angle of rotation for your rotation design.
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SYMMETRY A figure has symmetry if there exists a rigid motion β reflection, translation, rotation, or glide-reflection β
that maps the figure onto itself. One type of symmetry is line symmetry.
Line Symmetry Words Example
A figure in the plane has line symmetry if the figure can be mapped onto itself by a reflection in a line, called a
line of symmetry.
Examples
Use your compass and straightedge to draw one line of symmetry on each figure below that has at least one line of symmetry. Then, sketch any remaining lines of symmetry that exist.
a. b. c.
d. e. f.
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The number of times a figure maps onto itself as it rotates from 0Β° to 360Β° is called the order of symmetry.
The magnitude of symmetry (or angle of rotation) is the smallest angle through which a figure can be rotated so that it maps onto itself.
There is a special transformation that trivially maps any figure in the plane back to itself called the identity transformation. If we use π° to denote the identity transformation (π°(π·) = π· for every point π· in the plane), we can write this equivalency as follows:
πΉπ«,πππΛ (πΉπ«,πππΛ(πΉπ«,πππΛ(β³ π¨π©πͺ))) = π°(β³ π¨π©πͺ).
1. Find the center of rotation in equilateral triangle ABC . Next determine the angle of rotation.
Examples State whether the figure has non-trivial rotational symmetry. If so, locate the center of symmetry and state the order and magnitude of symmetry.
a. b. c.
Point Symmetry Words Example
A nontrivial rotational symmetry of a figure
is a rotation of the plane that maps the figure
back to itself such that the rotation is greater
than 0Λ but less than 360Λ.
A
B
C
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Use the figure to answer the questions below.
1. Draw all lines of symmetry. Locate the center of rotational symmetry.
2. Describe all symmetries explicitly.
a. What kinds are there?
b. How many are rotations? (Include a β360Λ rotational symmetry,β i.e., the identity symmetry.)
c. How many are reflections?
3. Prove that you have found all possible symmetries.
a. How many places can vertex π΄ be moved to by some symmetry of the square that you have identified? (Note that the vertex to which you move π΄ by some specific symmetry is known as the image of π΄ under that symmetry. Did you remember the identity symmetry?)
b. For a given symmetry, if you know the image of π΄, how many possibilities exist for the image of π΅?
c. Verify that there is symmetry for all possible images of π΄ and π΅.
d. Using part (b), count the number of possible images of π΄ and π΅. This is the total number of symmetries of the square. Does your answer match up with the sum of the numbers from Problem 2b and 2c?
A B
CD
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Use the figure to answer questions 4-6.
4. Draw all lines of symmetry. Locate the center of rotational symmetry.
5. Describe all symmetries explicitly.
a. What kinds are there?
b. How many are rotations (including the identity symmetry)?
c. How many are reflections?
6. Prove that you have found all possible symmetries.
a. How many places can vertex A be moved to by some symmetry of the pentagon? (Note that the vertex to which you move A by some specific symmetry is known as the image of A under that symmetry. Did you remember the identity symmetry?)
b. For a given symmetry, if you know the image of A, how many possibilities exist for the image of B?
c. Verify that there is symmetry for all possible images of A and B.
d. Using part (b), count the number of possible images of A and B. This is the total number of symmetries of the figure. Does your answer match up with the sum of the numbers from Problem 2b and 2c?
A
B
C
D
E
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TRANSLATIONS Construct the line parallel to a given line π΄π΅ through a given point π.
1. Draw circle πΆ1: Center π, radius π΄π΅. 2. Draw πΆ2: Center π΅, radius π΄π. 3. Label the intersection of C1 and C2 as π. 4. Draw ππ.
A translation is a transformation that moves all points of a figure the same distance in the same direction. Since vectors can be used to describe both distance and direction, vectors can be used to define translations.
For vector π΄π΅ββββ β, the translation along π΄π΅ββββ β is the transformation ππ΄π΅ββ ββ β of the plane defined as follows:
1. For any point π on the line πΏπ΄π΅, Tπ΄π΅ββ ββ β(π) is the point π on πΏπ΄π΅ so that ππββββ β has the same length and
the same direction as π΄π΅ββββ β, and
2. For any point π not on πΏπ΄π΅, Tπ΄π΅ββ ββ β(π) is the point π obtained as follows. Let πΏ1 be the line passing
through π and parallel to πΏπ΄π΅. Let πΏ2 be the line passing through π΅ and parallel to line πΏπ΄π. The point π is the intersection of πΏ1 and πΏ2.
lP
AB
A
B
P
In the figure to the right, quadrilateral π΄π΅πΆπ· has been translated the length and
direction of vector πΆπΆβ²ββββ ββ . Notice that the distance and direction from each vertex
to its corresponding vertex on the image are identical to that of πΆπΆβ²ββββ ββ .
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A
B
P1
P2P3
1. Draw the vector that defines each translation below.
a. b. c.
2. Use your compass and straightedge to apply ππ΄π΅ββ ββ β to segment π1π2.
3. Use your compass and straightedge to apply ππ΄π΅ββ ββ β to β³ π1π2π3.
P1
P2
A B
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P1
P2
AB
AB
4. Use your compass and straightedge to apply ππ΄π΅ββ ββ β to the circle below (center π1, radius π1π2).
5. Use your compass and straightedge to apply ππ΄π΅ββ ββ β to the circle below.
Hint: You will need to first find the center of the circle.
Lesson Summary
A translation maps segments onto segments of equal length.
A translation maps angles onto angles of equal measure.