Transcript Presentation Title

Dec. 14
HW 18: Transformations
Aim:
Working with Dilation, Reflection,
Translations, and Rotations. Review from 7th
Accelerated.
Materials you will need for this homework:
• pencil
• ruler
Definitions:
A transformation in geometry is defined as when an
object (shape) undergoes a change in position or
size.
There are 4 types of transformation:
1.
2.
3.
4.
Dilation
Reflection
Rotation
Translation
2. Dilation
A dilation is a type of transformation that changes
the size of the image but the image is the same
shape. To make an image smaller or larger you
multiply by the scale factor. The orientation of the
image is the same as the original figure.
Try out the example on the website
3. Reflection
A reflection is a kind of transformation. It is basically
a “flip” of a shape over the line of reflection to create
a mirror image of the shape. The image is
congruent to the original shape. Orientation of the
image is different from the original figure.
y
Ex1. Dilate triangle
ABC, with vertices
A(1, -3), B(3, -3), and
C(2, -1) with a scale
factor of 2. Then write
the new coordinates as
A’B’C’ in the table
below.
D2
(2, -6)
A(1, -3)  A’ ________
(6, -6)
B(3, -3)  B’________
6
Notice that the base and height of the
triangles are in the ratio of 1:2
3
x
-6
-3
C

(6, -2)
C(2, -1)  C’________
6
C’

-3

B
A
D2
Rule for Dilation:
Multiply each coordinate (x, y) by the scale
factor.
3
-6

A’
B’
y
Ex2. Dilate rectangle
BRAT, with vertices
B(-6, 6), R(3, 6),
A(3, 3), and T(-6, 3)
with a scale factor of
1
. Then
write the new
3
coordinates as B’R’A’T’
in the table below.
B
R


6

T

A
3
B’
R’


A’

 D
1
3
D1
3
(-2, 2)
B(-6, 6)  B’ ________
(1, 2)
R(3, 6)  R’_________
T’
-6
-3
3
Notice that the lengths and widths are
in the ratio of 1:3
-3
(1, 1)
A(3, 3)  A’_________
(-2, 1)
T(-6, 3)  T’_________
x
-6
6
Ex3. Reflect Trapezoid MATH over the x-axis and label it M’A’T’H’
y
Rover x
(1, -6)
M(1, 6)  M’_________
4
A(3, 6) 
(3, -6)
A’_________
2
T(5, 2) 
(5, -2)
T’_________
H(1, 2) 
A
6M
-6
-4
-2
2
-2
(1, -2)
H’_________
-6
(x,y)  (x, -y)
When reflecting over the x-axis, keep the x coordinate the
same and negate the y coordinate.
4
H
’
M’
6
T’
Rx
-4
Rule (Reflecting over the x-axis):
T
H
A’
x
Ex4. Reflect Trapezoid MATH over the y-axis and label it M’’A’’T’’H’’
y
A”
Rover y
M(1, 6) 
M”_________
(-1, 6)
(-3, 6)
A(3, 6)  A”
(-5, 2)
_________
T(5, 2) 
4
Ry
T”
-6
H
”
-4
2
-2
2
-4
H(1, 2)  H”_________
-6
Rule (Reflecting over the y-axis):
T
H
-2
(-1, 2)
T”_________
A
M” 6 M
(x,y)  (-x, y)
When reflecting over the y-axis, negate the x coordinate and
keep the y coordinate the same.
4
6
x
4. A translation is another type of transformation. It is
the same as sliding/shifting an object. The notation
for translate is T(a, b) where a and b represent how
much you slide in the x and the y directions,
respectively. The shape still looks exactly the same,
just in a different place.
5. Another type of transformation is a rotation. A
rotation turns a figure 90 degrees or 180 degrees
clockwise or counterclockwise. The image is
congruent to the original figure. Orientation of the
image is different from the original figure.
Translations
Example 4
Translate Triangle SUN
5 units right and 3 units
down and label the new
coordinates.
There’s 2 ways to translate a figure. Using the graph
y
or the notation.
S

6
4
S’
T5, -3
S(-6, 6) 
S’(-1, 3)
_________
U(-6, 2) 
U’(-1, 1)
_________
+5 -3
+5 -3
N(-2, 2) 
+5 -3
U
Notation for example problem:
T5, -3 means 5 units right and 3 units
down and label the new coordinates.
2
x
-6
N’(3, -1)
_________
N
-4
-2
2
4
N’
U’
-2
-4
-6
6
y
Example 5
a.) Translate Trapezoid
BIRD 2 units left and 6
units up and label the
new figure B’I’R’D’.
b.) Write the translation
above using proper
notation.
6
R’
3
T-2, 6
B’(4, 0)
B(6, -6)  __________
-2 +6
I(1, -6) 
-2 +6
D’
I’
-6
-3
3
I’(-1, 0)
__________
R’(0, 4)
R(2, -2)  __________
-2 +6
x
B’
R

6
D

-3
D’(2,
D(4, -2)  __________
-2 +6
4)
-6
I
B
Example 6.
What type of transformation is
shown in the diagram?
Explain your answer.
The type of transformation
that is shown in the
diagram is a translation.
The object is being
translated 7 units to the
right and 5 units down.
y
A Rotation of 90° counterclockwise about (0,0)
What did you noticed
about the points in the
original triangle and the
new triangle?
8
7
6
5
(2, 1)  ______
(-1, 2)
4
3
(-2, 4)
(4, 2)  ______
2
(-5, 3)
(3, 5)  ______
1
–6
–5
–4
–3
–2
x
x
–7
–1
1
2
3
4
5
6
7
8
x
-1
-2
-3
-4
-5
-6
Rule for rotation of 90°
Notation:
(x, y)  (-y, x)
_______________________
Negate
the y-coordinate
_______________________
then
flip the x and y
_______________________
coordinates.
_______________________
y

O’
O
T



6
Ex7. Rotate 90
counterclockwise
Parallelogram NHOT with
vertices: N(1, 2), H(5, 2),
O(6, 6), T(2, 6)
H’
R90
3
N


T’

H
N’
-6
x
-3
3
-3
-6
N(1, 2) 
N’(-2, 1)
________
H(5, 2) 
H’(-2, 5)
_________
O(6, 6) 
O’(-6, 6)
_________
T(2, 6) 
T’(-6, 2)
_________
6
y
A Rotation of 180° about (0,0)
What did you noticed
about the points in the
original triangle and the
new triangle?
8
7
6
5
(2, 1)  ______
(-2, -1)
4
3
(-4, -2)
(4, 2)  ______
2
(-3, -5)
(3, 5)  ______
1
–6
–5
–4
–3
–2
x
–7
x
–1
1
2
3
4
5
6
7
8
x
-1
-2
-3
-4
-5
-6
Rule for rotation of 180°
Notation:
(x, y)  (-x, -y)
____________________
Negate
the x-coordinate
____________________
and
y-coordinate.
____________________
____________________
y
Ex8. Rotate 180
counterclockwise
Trapezoid BIRD.
6
D’
R’

R180
3

B’(-6, 6)
________
I(1, -6) 
I’(-1, 6)
_________
R(2, -2) 
R’(-2, 2)
_________
D(4, -2) 
D’(-4, 2)
_________
R’

I’
x
-6
B(6, -6) 
-3
3
R

6
I

-3
-6

D
R
y
Ex9. What transformation is
shown on the grid? Explain
your answer.
H’
E


6
The transformation that is
shown on the grid is a
rotation of 90 degrees
clockwise.
H

3
Y
-6
Y’
-3
 E’

3
-3
-6
6
x