Transcript 4-3 Using Matrices to Transform Geometric Figures

Using
Matrices
toto
Transform
Using
Matrices
Transform
4-3
4-3 Geometric
Figures
Geometric
Figures
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
4-3
Using Matrices to Transform
Geometric Figures
Warm Up
Perform the indicated operation.
1.
2.
3.
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Objective
Use matrices to transform a plane
figure.
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Vocabulary
translation matrix
reflection matrix
rotation matrix
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
You can describe the position, shape, and size of a
polygon on a coordinate plane by naming the
ordered pairs that define its vertices.
The coordinates of ΔABC below are A (–2, –1),
B (0, 3), and C (1, –2) .
You can also define ΔABC by a matrix:
 x-coordinates
 y-coordinates
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
A translation matrix is a matrix used to
translate coordinates on the coordinate plane.
The matrix sum of a preimage and a translation
matrix gives the coordinates of the translated
image.
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Reading Math
The prefix pre- means “before,” so the preimage
is the original figure before any transformations
are applied. The image is the resulting figure
after a transformation.
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Example 1: Using Matrices to Translate a Figure
Translate ΔABC with coordinates A(–2, 1),
B(3, 2), and C(0, –3), 3 units left and 4 units
up. Find the coordinates of the vertices of
the image, and graph.
The translation
matrix will have –3
in all entries in row
1 and 4 in all entries
in row 2.
Holt McDougal Algebra 2
 x-coordinates
 y-coordinates
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Using Matrices to Transform
Geometric Figures
Example 1 Continued
A'B'C', the image of
ABC, has coordinates
A'(–5, 5), B'(0, 6), and
C'(–3, 1).
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Check It Out! Example 1
Translate ΔGHJ with coordinates G(2, 4), H(3,
1), and J(1, –1) 3 units right and 1 unit down.
Find the coordinates of the vertices of the image
and graph.
The translation
matrix will have 3 in
all entries in row 1
and –1 in all entries
in row 2.
Holt McDougal Algebra 2
 x-coordinates
 y-coordinates
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Using Matrices to Transform
Geometric Figures
Check It Out! Example 1 Continued
G'H'J', the image of
GHJ, has coordinates
G'(5, 3), H'(6, 0), and
J'(4, –2).
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
A dilation is a transformation that scales—enlarges
or reduces—the preimage, resulting in similar
figures. Remember that for similar figures, the
shape is the same but the size may be different.
Angles are congruent, and side lengths are
proportional.
When the center of dilation is the origin,
multiplying the coordinate matrix by a scalar gives
the coordinates of the dilated image. In this
lesson, all dilations assume that the origin is the
center of dilation.
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Using Matrices to Transform
Geometric Figures
Example 2: Using Matrices to Enlarge a Figure
Enlarge ΔABC with coordinates A(2, 3),
B(1, –2), and C(–3, 1), by a factor of 2.
Find the coordinates of the vertices of the
image, and graph.
Multiply each coordinate by 2 by multiplying each
entry by 2.
 x-coordinates
 y-coordinates
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Using Matrices to Transform
Geometric Figures
Example 2 Continued
A'B'C', the image of
ABC, has coordinates
A'(4, 6), B'(2, –4),
and C'(–6, 2).
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Check It Out! Example 2
Enlarge ΔDEF with coordinates D(2, 3), E(5,
1), and F(–2, –7) a factor of . Find the
coordinates of the vertices of the image, and
graph.
Multiply each coordinate by
entry by .
Holt McDougal Algebra 2
by multiplying each
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Using Matrices to Transform
Geometric Figures
Check It Out! Example 2 Continued
D'E'F', the image of
DEF, has coordinates
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
A reflection matrix is a matrix that creates a
mirror image by reflecting each vertex over a
specified line of symmetry. To reflect a figure
across the y-axis, multiply
by the coordinate matrix. This reverses the xcoordinates and keeps the y-coordinates
unchanged.
Holt McDougal Algebra 2
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Using Matrices to Transform
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Caution
Matrix multiplication is not commutative. So be
sure to keep the transformation matrix on the
left!
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Using Matrices to Transform
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Example 3: Using Matrices to Reflect a Figure
Reflect ΔPQR with coordinates P(2, 2),
Q(2, –1), and R(4, 3) across the y-axis.
Find the coordinates of the vertices of the
image, and graph.
Each x-coordinate is multiplied by –1.
Each y-coordinate is multiplied by 1.
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Example 3 Continued
The coordinates of the vertices of the image are
P'(–2, 2), Q'(–2, –1), and R'(–4, 3).
Holt McDougal Algebra 2
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Using Matrices to Transform
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Check It Out! Example 3
To reflect a figure across the x-axis, multiply by
.
Reflect ΔJKL with coordinates J(3, 4), K(4, 2),
and L(1, –2) across the x-axis. Find the
coordinates of the vertices of the image and
graph.
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Check It Out! Example 3
The coordinates of the vertices of the image
are J'(3, –4), K'(4, –2), L'(1, 2).
Holt McDougal Algebra 2
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A rotation matrix is a matrix used to rotate a
figure. Example 4 gives several types of rotation
matrices.
Holt McDougal Algebra 2
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Using Matrices to Transform
Geometric Figures
Example 4: Using Matrices to Rotate a Figure
Use each matrix to rotate polygon ABCD
with coordinates A(0, 1), B(2, –4), C(5, 1),
and D(2, 3) about the origin. Graph and
describe the image.
A.
The image A'B'C'D' is rotated 90° counterclockwise.
B.
The image A''B''C''D'' is rotated 90° clockwise.
Holt McDougal Algebra 2
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Example 4 Continued
Holt McDougal Algebra 2
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Using Matrices to Transform
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Check It Out! Example 4
Use
Rotate ΔABC with coordinates A(0, 0),
B(4, 0), and C(0, –3) about the origin.
Graph and describe the image.
A'(0, 0), B'(-4, 0), C'(0, 3); the image is rotated
180°.
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Check It Out! Example 4 Continued
Holt McDougal Algebra 2
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Using Matrices to Transform
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Lesson Quiz
Transform triangle PQR with vertices
P(–1, –1), Q(3, 1), R(0, 3). For each, show
the matrix transformation and state the
vertices of the image.
1. Translation 3 units to the left and 2 units up.
2. Dilation by a factor of 1.5.
3. Reflection across the x-axis.
4. 90° rotation, clockwise.
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Using Matrices to Transform
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Lesson Quiz
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Holt McDougal Algebra 2
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