Geometry
4.1 Translations
4.1 Warm Up
Translate point P. State the coordinates of P'.
1. P(-4, 4); 2 units down, 2 units right
2. P(-3, -2); 3 units right, 3 units up
3. P(2,2); 2 units down, 2 units right
4. P(-1, 4); 3 units left, 1 unit up
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4.1 Essential Question
How can you translate a figure in a coordinate
plane?
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Goals

Identify and use translations in the plane.
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Translation
A translation is a transformation that maps every two points
P and Q to points P’ and Q’ so that the following properties
are true:
P’
 1. PP’ = QQ’
P
 2. PP’ || QQ’
Q’
 A Translation is an isometry.
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Q
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Postulate 4.1
Postulate 4.1 Translation Postulate
A translation is a rigid motion.
A rigid motion preserves length and angle measure, so it is
an isometry. The following statements are true for the
translation shown.
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4.1 Translations
Translations in the Coordinate Plane
Translation Mapping
Formula:
 (x, y)  (x + a, y + b)
-a
 a and b are constants.
 (x + a, y + b) shifts a
units horizontally.
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+a
Translations in the Coordinate Plane
Translation Mapping
Formula:
 (x, y)  (x + a, y + b)
-a
 a and b are constants.
-b
 (x + a, y + b) shifts a
units horizontally.
 (x + a, y + b) shifts b
units vertically.
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+b
+a
4.1 Translations
Translations in the Coordinate Plane






a > 0  shift right
(x, y)  (x + 3, y)
a < 0  shift left
(x, y)  (x – 2, y)
b > 0  shift up
(x, y)  (x, y + 9)
b < 0  shift down
(x, y)  (x, y – 5)
a = 0  no horizontal shift (x, y)  (x, y + 4)
b= 0  no vertical shift
(x, y)  (x + 7, y)
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Use coordinate notation to describe the
translation.






A point is translated 10 units to the right and 8 units up.
(x, y)  (x + 10, y + 8)
A point is translated 7 units to the left, and 2 units up.
(x, y)  (x – 7, y + 2)
A point is translated 1 unit to the left, and 2 units down.
(x, y)  (x – 1, y – 2)
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Example 1
Translate RS using:
(x, y)  (x + 3, y + 2)
R(1, 2)  (1 + 3, 2 + 2)
R
S
3
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Example 1
Translate RS using:
(x, y)  (x + 3, y + 2)
R(1, 2)  (1 + 3, 2 + 2)
R
S
3
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2
Example 1
Translate RS using:
(x, y)  (x + 3, y + 2)
R(1, 2)  (1 + 3, 2 + 2)
R’(4, 4)
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R’(4, 4)
S
R
4.1 Translations
Example 1
Translate RS using:
(x, y)  (x + 3, y + 2)
R(1, 2)  (1 + 3, 2 + 2)
R’(4, 4)
R’(4, 4)
S
R
S(3, 3)  (3 + 3, 3 + 2)
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Example 1
Translate RS using:
(x, y)  (x + 3, y + 2)
R(1, 2)  (1 + 3, 2 + 2)
R’(4, 4)
S’(6, 5)
R’(4, 4)
S
R
S(3, 3)  (3 + 3, 3 + 2)
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Example 1
Translate RS using:
(x, y)  (x + 3, y + 2)
R(1, 2)  (1 + 3, 2 + 2)
R’(4, 4)
S’(6, 5)
R’(4, 4)
S
R
S(3, 3)  (3 + 3, 3 + 2)
S’(6, 5)
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Example 2
Given ABC with:
A(-2, 5), B(-4, 1), C(0, 0)
Sketch the image after translation
(x, y)  (x + 4, y – 4)
A
B
C
A(-2, 5)  A’(2, 1)
B(-4, 1)  B’(0, -3)
C(0, 0)  C’(4, -4)
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A’
B’
C’
4.1 Translations
Identify the Translation Rule
Write the rule that translates (-4, -2)
to (1, 4).
To move from (-4, -2) to (1, 4) go to
the right 5 and up 6.
So the translation rule is:
(x, y)  (x + 5, y + 6).
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4.1 Translations
Example 3
What rule translates (4, 2) to (– 3, –1)?
(x, y) → (x – 7, y – 3)
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Example 4
Complete the statement.
If (4, 0) maps onto (5, 1), then (6, 8) will map
(7, 9)
onto_______.
Why?
 (4 + 1, 0 + 1)  (5, 1)
 So (6 + 1, 8 + 1)  (7, 9)
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Your Turn
Complete the statement.
If (7, 8) maps onto (4, 10), then (20, 5) will map
(17, 7)
onto_______.
Why?
 (7 – 3, 8 + 2)  (4, 10)
 So (20 – 3, 5 + 2)  (17, 7)
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Example 5
B’
Write the rule used to
translate ABCD.
(x, y) → (x + 6, y + 4)
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B
A
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C
D
A’
C’
D’
Example 6
Translate JKL using the notation (x, y) → (x – 4, y +3).
L’
Notice: the rays
drawn from
each point to
its image are
parallel.
L
J’
K’
K
J
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Example 7
Translate ABC by
(x, y) → (x + 4, y – 3).
 A(-4, 8)
 B(-10, -2)
 C(-22, 15)
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



A’B’C’
A’(0, 5)
B’(-6, -5)
C’(-18, 12)
Performing Compositions
Theorem 4.1 Composition Theorem
The composition of two (or more) rigid motions is a rigid motion.
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Example 8
Graph 𝑅𝑆 with endpoints R(−8, 5) and S(−6, 8) and its
image after the composition.
S’
Translation: (x, y) → (x + 5, y − 2)
Translation: (x, y) → (x − 4, y − 2)
R
S’’
R’
Write a single translation rule for the composition.
(x, y) → (x + 1, y − 4).
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S
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R’’
Summary



A translation shifts a figure horizontally or vertically.
A translation is an isometry.
Translation formula: (x, y)  (x + a, y + b)
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Assignment
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