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Transformations

An operation that moves or changes a
geometric figure (a preimage) in some way to
produce a new figure (an image).
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changes the position of the figure without
changing the size or shape.
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moves every point of a figure the same
distance in the same direction.
Coordinate notation: (x , y)
(x + a, y + b)
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The vertices of ABC are A(4, 4), B(6, 6), and
C(7, 4). The notation (x, y) → (x + 1, y – 3)
describes the translation of
ABC to
DEF.
What are the vertices of
DEF?
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Uses a line of reflection to create a mirror
image of the original figure.
Coordinate notation for reflection in the
x-axis : (x ,y) (x , -y)
Coordinate notation for reflection in the
y- axis: (x , y) (-x, y)

Reflect a figure in the x-axis

Turns a figure about a fixed point called the
center of rotation
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Graph AB and CD. Tell whether CD is a rotation
of AB about the origin. If so, give the angle and
direction of rotation.
A(–3, 1), B(–1, 3), C(1, 3), D(3, 1)

Tell whether PQR is a rotation of
STR. If
so, give the angle and direction of rotation.

Name the type of transformation demonstrated
in each picture.
a.
b.

Name the type of transformation shown.
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A transformation that stretches or shrinks a
figure to create a similar figure.
A figure is reduced or enlarged with respect
to a fixed point called the center of dilation.
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The scale factor of a dilation is the ratio of
the side length of the image to the
corresponding side length of the original
figure
Coordinate notation for a dilation with
respect to the origin: (x ,y)
( kx, ky)
Reduction: 0 < k < 1
Enlargement : k > 1

Draw a dilation of quadrilateral ABCD with
vertices A(2, 1), B(4, 1), C(4, – 1), and D(1, – 1).
Use a scale factor of 2.
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Translation Theorem: A translation is an
isometry.
Isometry- a congruence transformation
Preimage- original figure
Image- new figure

Write a rule for the translation of ABC to
A′B′C′. Then verify that the transformation
is an isometry.

Name the vector and write its component form.
a.

The vertices of ∆LMN are L(2, 2), M(5, 3), and
N(9, 1). Translate ∆LMN using the vector –2, 6.

A boat heads out from point A on one island
toward point D on another. The boat
encounters a storm at B, 12 miles east and 4
miles north of its starting point. The storm
pushes the boat off course to point C, as
shown.
Write the component form of AB, BC, and CD.
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Matrix- a rectangular arrangement of
numbers in rows and columns
Element- each number in the matrix
Dimensions- row x column
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A reflection in a line (m) maps every point (P)
in the plane to a point (P`) so that for each
point, one of the following is true:
m
P
m
P
P`
P`
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If (a,b) is reflected in the x-axis, its image is (a,-b).
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If (a,b) is reflected in the y-axis, its image is (-a,b).
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If (a,b) is reflected in the line y = x,
its image is (b,a).
If (a,b) is reflected in the line y = -x,
its image is (-b,-a).
6
4
2
-10
-5
5
-2
-4
-6
10

You and a friend are meeting on the beach
shoreline. Where should you meet to
minimize the distance you must both walk?
6
4
2
-10
-5
5
-2
-4
-6
10
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Find the reflection of PQR in the x- axis
using in matrix multiplication.
P(-3,6) Q(-5,3) R(-1,2)
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A rotation is an isometry
Center of rotation- a fixed point in which a
figure is turned about
Angle of Rotation- the angle formed from
rays drawn from the center of rotation to a
point and its image
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These rules apply for counterclockwise
rotations about the origin
a 90o rotation (a,b)
a 180o rotation (a,b)
a 270o rotation (a,b)
(-b,a)
(-a,-b)
(b,-a)
6
4
2
-10
5
-5
-2
-4
-6
10
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Composition of Transformation- 2 or more
transformations are combined to form a
single transformation
The composition of 2 (or more) isometries is
an isometry.
6
4
2
-10
-5
5
-2
-4
-6
10
6
4
2
-10
-5
5
-2
-4
-6
10
k
m
k
m
50 