9.1 – translate figures and use vectors. transformation: moves or changes a figure preimage:...
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9.1 – Translate Figures and Use Vectors
Transformation: Moves or changes a figure
Preimage: Original figure
Image: Transformed figure
Isometry: A congruent transformation
“P prime”Point becomes 'P P
Translation:
An isometry that moves every point a certain distance in a certain direction
P
Q
'P
'Q
Translation:
Note: ' 'PP QQ and ' 'PP QQ≅
Motion Rule:
Moves each point left, right, down, or up
( )( , ) #, #x y x y→ + +
Left or
Right
Downor Up
Use the translation ( )( , ) 2, 5x y x y→ + −
What is the image of D(4, 7)?
D (4 + 2, 7 – 5)
(6, 2)
'D =
'D =
'D
+2
–5
Use the translation ( )( , ) 7, 4x y x y→ − +
What is the image of R(2, –4)?
R
(2 – 7, –4 + 4)
(–5, 0)
'R =
'R =
'R
–7
+4
Use the translation ( )( , ) 4, 6x y x y→ + −
What is the preimage of (–5, 3)?
M(–5 – 4, 3 + 6)
(–9, 9)
M =
M =
'M–4
+6
'M
Use the translation ( )( , ) 5, x y x y→ −
What is the preimage of (4, –1)?
A
(4 + 5, –1)
(9, –1)
A=
A=
'A +5
'A
The vertices of ABC are A(–1, 1), B(4, –1), and C(2, 4). Graph the image of the triangle using prime notation.
( )( , ) 3, 5x y x y→ − +
A
B
C
'C
'A'B
(–1 – 3, 1 + 5)
(–4, 6)
'A =
'A =
(4 – 3, –1 + 5)
(1, 4)
'B =
'B =
(2 – 3, 4 + 5)
(–1, 9)
'C =
'C =
The vertices of ABC are A(–1, 1), B(4, –1), and C(2, 4). Graph the image of the triangle using prime notation.
( )( , ) , 3x y x y→ −
A
B
C
'C
'A'B
(–1 , 1 – 3)
(–1, –2)
'A =
'A =
(4, –1 – 3)
(4, –4)
'B =
'B =
(2, 4 – 3)
(2, 1)
'C =
'C =
is the image of ABC after a translation. Write a rule for the translation.
' ' 'A B C
( )( , ) 5, 3x y x y→ − +
–5
+3
is the image of ABC after a translation. Write a rule for the translation.
' ' 'A B C
( )( , ) 2, 5x y x y→ + −
+2
–5
Vector:
Translates a shape in direction and magnitude, or size.
FGWritten:
Where F is the initial point and G is the terminal point.
Component form:
Vector:
< x, y >
5, 3
Name the vector and write its component form.
5, 1−
+5–1
JD
Name the vector and write its component form.
7, 3− −
–7
–3
DR
Name the vector and write its component form.
0, 4−
–4
RS
Use the point P(5, –2). Find the component form of the vector that describes the translation to '.P
P
'(2, 0)P
'P
–3+2
3, 2−
Use the point P(5, –2). Find the component form of the vector that describes the translation to '.P
P'P
–10–2
10, 2− −
'( 5, 4)P − −
Find the value of each variable in the translation.
a = 80°
2b = 8b = 4
5d = 100d = 20°
c = 13
Find the value of each variable in the translation.
a = 180 – 90 – 31
a = 59°
b – 5 = 12b = 17
3c + 2 = 203c = 18
c = 6
9.3 – Perform Reflections
Reflection:
Transformation that uses a line like a mirror to reflect an image
Line of Reflection:
Mirror line in a reflection
A reflection in a line m maps every point P in the plane to a point , such that:
• If P is not on m, then m is the perpendicular bisector of
• If P is on m, then
'.PP
'P
'P P=
P
Reflect point P(5, 7) in the given line.
'P
P(5, 7) becomes'(5, 7)P −
A reflection in the x-axis changes (x, y) into _______(x, –y)
x – axis
P
Reflect point P(5, 7) in the given line.
'P
P(5, 7) becomes'( 5, 7)P −
A reflection in the y-axis changes (x, y) into _______(–x, y)
y – axis
P
Reflect point P(5, 7) in the given line.
'P
P(5, 7) becomes'(7, 5)P
A reflection in the y = x changes (x, y) into _______(y, x)
y = x
Graph the reflection of the polygon in the given line.
x – axis
'C
'A
'B
Graph the reflection of the polygon in the given line.
y – axis
'C
'A
'B
'D
'C
'A
'B
Graph the reflection of the polygon in the given line.
y = x
(–1 , –3)
(–3, –1)
A=
'A =
(2, –4 )
(–4, 2)
B =
'B =
(3, 0)
(0, 3)
C =
'C =
Graph the reflection of the polygon in the given line.
x – axis
'C
'A
'B
'D
Graph the reflection of the polygon in the given line.
x = –1
'C'A
'B
Graph the reflection of the polygon in the given line.
y = 2'C
'A =
'B
'D=
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