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TEKS: G2B, G10A, G5C
The student will make conjectures about
angles, lines and polygons using a variety of
approaches including transformations.
The student will use congruence
transformations to make conjectures and
justify properties of figures.
The student will use prosperities of
transformations to make connections
between math and the real world, such as
tessellations.
A tessellation, or tiling, is a repeating pattern
that completely covers a plane with no gaps
or overlaps. The measures of the angles that
meet at each vertex must add up to 360°.
In the tessellation shown,
each angle of the
quadrilateral occurs once
at each vertex. Because the
angle measures of any
quadrilateral add to 360°,
any quadrilateral can be
used to tessellate the
plane. Four copies of the
quadrilateral meet at each
vertex.
The angle measures of any triangle add up to 180°.
This means that any triangle can be used to
tessellate a plane. Six copies of the triangle meet at
each vertex as shown.
Example: 1
Reflect the rectangle with vertices A(-2, 4),
B(3, 4), C(3, 3) and D(–2, 3) across the y=1 line.
A
B
D
C
D’
C’
A’
B’
A(-2, 4)
A (-2, –2)
B(3, 4)
B ’(3, –2)
C(3, 3)
C’(3, –1)
D(–2, 3)
D ’(–2, –1)
Graph the image and preimage.
Example: 2
Reflect the rectangle with vertices A-2, 4),
B(3, 4), and C(–2, 3) across the x=-1 line.
A’
A
C
B
B’
C’
A(-5, 4)
A’ (3, 4)
B(-3, 2)
B ’(1, 2)
C(-5, 2)
C’(3, 2)
Graph the image and preimage.
REVIEW of Rules
Reflection: x axis (x, y) → (x, - y)
y axis (x, y) → (- x, y)
y = x (x, y) → (y, x)
Rotation: 90° counterclockwise about the origin (x, y) → (- y, x)
180° counterclockwise about the origin (x, y) → (- x,- y)
Translation: (x, y) → (x + a, y + b)
Dilation: (x, y) → (kx, ky)
REVIEW
A transformation is a change in the
position, size, or shape of a figure. The
original figure is called the preimage. The
resulting figure is called the image. A
transformation maps the preimage to the
image. Arrow notation () is used to
describe a transformation, and primes (’)
are used to label the image.
Example: 3
Identify each transformation. Then use arrow notation to
describe the transformation.
a.
translation; MNOP  M’N’O’P’
b.
rotation; ∆XYZ  ∆X’Y’Z’
Example: 4
Identify the transformation.
Reflection across the x axis,
DEFG  D’E’F’G’
Example: 5
Reflect the figure with the given vertices across the given line.
X(2, –1), Y(–4, –3), Z(3, 2); y-axis
The reflection of (x, y) is (-x, y).
Z’
Z
X’
Y
X(2,–1)
X’ (-2, -1)
Y(–4,–3)
Y’ (4, -3)
Z(3, 2)
X
Y’
Z ’(-3, 2)
Graph the image and preimage.
Example: 6
Reflect the figure with the given vertices across the given line.
R(–2, 2), S(5, 0), T(3, –1); y = x
S’
R
The reflection of (x, y) is (y, x).
T’
S
R’
T
R(–2, 2)
R’ (2, –2)
S (5, 0)
S’ (0, 5)
T(3, –1)
T ’(–1, 3)
Graph the image and preimage.
Example: 7
Reflect the rectangle with vertices S(3, 4),
T(3, 1), U(–2, 1) and V(–2, 4) across the x-axis.
V
U
U’
V’
S
T
T’
S’
The reflection of (x, y) is (x,–y).
S(3, 4)
S’ (3, –4)
T(3, 1)
T ’(3, –1)
U(–2, 1)
U ’(–2, –1)
V(–2, 4)
V ’(–2, –4)
Graph the image and preimage.
To find coordinates for the image of a figure in
a translation, add a to the x-coordinates of
the preimage and add b to the y-coordinates
of the preimage.
Translations can also be described by a rule
such as (x, y)  (x + a, y + b).
Example: 8
Find the coordinates for the image of ∆ABC after the
translation (x, y)  (x + 2, y - 1). Draw the pre image and
image.
Step 1 Find the coordinates of
∆ABC.
The vertices of ∆ABC are A(–4, 2), B(–3, 4),
C(–1, 1).
Step 2 Apply the rule to find the vertices of
the image.
A’(–4 + 2, 2 – 1) = A’(–2, 1)
B’(–3 + 2, 4 – 1) = B’(–1, 3)
C’(–1 + 2, 1 – 1) = C’(1, 0)
Step 3 Plot the points. Then finish drawing the image by
using a straightedge to connect the vertices.
Example: 9
Find the coordinates for the image of JKLM after the translation
J’
K’
(x, y)  (x – 2, y + 4). Draw the image.
Step 1 Find the coordinates of
JKLM.
The vertices of JKLM are J(1, 1), K(3, 1),
L(3, –4), M(1, –4), .
M’
Step 2 Apply the rule to find the vertices of
the image.
J’(1 – 2, 1 + 4) = J’(–1, 5)
K’(3 – 2, 1 + 4) = K’(1, 5)
L’(3 – 2, –4 + 4) = L’(1, 0)
M’(1 – 2, –4 + 4) = M’(–1, 0)
Step 3 Plot the points. Then finish drawing the image by
using a straightedge to connect the vertices.
L’
Example: 10
The figure shows part of a tile floor. Write a rule for the
translation of hexagon 1 to hexagon 2.
Step 1 Choose two points.
Choose a Point A on the preimage and a
corresponding Point A’ on the image. A
has coordinate (2, –1) and A’ has
A’
coordinates
Step 2 Translate.
To translate A to A’, 2 units are subtracted
from the x-coordinate and 1 units are
added to the y-coordinate. Therefore, the
translation rule is (x, y) → (x – 3, y + 1 ).
A
Example: 11
Use the diagram to write a rule for the translation of
square 1 to square 3.
Step 1 Choose two points.
Choose a Point A on the preimage
and a corresponding Point A’ on the
image. A has coordinate (3, 1) and A’
has coordinates (–1, –3).
Step 2 Translate.
To translate A to A’, 4 units are
subtracted from the x-coordinate and 4
units are subtracted from the
y-coordinate. Therefore, the translation
rule is (x, y)  (x – 4, y – 4).
A’
Counterclockwise
Example: 12
Describe the transformation.
90° rotation counterclockwise,
∆ABC  ∆A’B’C’
Example: 13
Rotate ∆ABC by 90° about the origin.
B’
The rotation of (x, y) is (–y, x).
A(2, –1)
A’ (1, 2)
B(4, 1)
B’ (–1, 4)
C(3, 3)
C’ (–3, 3)
C’
C
A’
B
A
Graph the preimage and image.
Example: 14
Rotate ΔJKL with vertices J(3, 5), K(4, –5), and L(–1, 6) by 180°
about the origin.
The rotation of (x, y) is (–x, –y).
J(3, 5)
K(4, –5)
L(–1, 6)
L
K’
J
J’ (–3, -5)
K’ (–4, 5)
L’ (1, –6)
Graph the preimage and image.
J’
L’
K
A dilation is a transformation that changes
the size of a figure but not its shape. The
preimage and the image are always similar.
A scale factor describes how much the figure
is enlarged or reduced.
For a dilation with scale factor k, you can find
the image of a point by multiplying each
coordinate by k: (a, b)  (ka, kb).
Helpful Hint
If the scale factor of a dilation is greater
than 1 (k > 1), it is an enlargement. The
scale factor in fraction form is an
improper fraction.
If the scale factor is less than 1 (k < 1),
it is a reduction. The scale factor in
fraction form is a proper fraction.
Example: 15
Draw the border of the photo after a dilation with
scale factor
Example: 15 cont.
Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and
D(3, 0) by
Rectangle
ABCD
Rectangle
A’B’C’D’
Example: 15 cont.
Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0).
Draw the rectangle.
Example: 16
Given that ∆TUO ~ ∆RSO, find the
coordinates of U and the scale factor.
Since ∆TUO ~ ∆RSO,
Substitute 12 for RO,
9 for TO, and 16 for OY.
12OU = 144
OU = 12
Cross Products Prop.
Divide both sides by 12.
Example: 16cont.
U lies on the y-axis, so its x-coordinate is 0. Since OU = 12, its ycoordinate must be 12. The coordinates of U are (0, 12).
So the scale factor is