12.3: Geometric Translations
What if Lucy lived in San Francisco,
a) The component form of
b) Lucy’s parents are considering moving to Fresno,
c) Is Ukiah or Paso Robles closer to Fresno?
After completing this Concept, you'll be able to answer these questions.
Watch This
CK12 Foundation: Chapter12TranslationsA
Learn more about translations by watching the video at this link.
Guidance
A transformation is an operation that moves, flips, or changes a figure to create a new figure. A rigid transformation is a transformation that preserves size and shape. The rigid transformations are: translations (discussed here), reflections, and rotations. The new figure created by a transformation is called the image. The original figure is called the preimage. Another word for a rigid transformation is an isometry. Rigid transformations are also called congruence transformations. If the preimage is
A translation is a transformation that moves every point in a figure the same distance in the same direction. In the coordinate plane, we say that a translation moves a figure
In the graph below, the line from
The component form of
Example A
Graph square
The translation notation tells us that we are going to move the square to the left 2 and up 3.
Example B
Name the vector and write its component form.
The vector is
Example C
Name the vector and write its component form.
The vector is
Watch this video for help with the Examples above.
CK12 Foundation: Chapter12TranslationsB
Concept Problem Revisited
a)
b)
c) You can plug the vector components into the Pythagorean Theorem to find the distances. Paso Robles is closer to Fresno than Ukiah.
Vocabulary
A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation) is a transformation that does not change the size or shape of a figure. The new figure created by a transformation is called the image. The original figure is called the preimage. A translation is a transformation that moves every point in a figure the same distance in the same direction. A vector is a quantity that has direction and size. The component form of a vector combines the horizontal distance traveled and the vertical distance traveled.
Guided Practice
1. Find the translation rule for
2. Draw the vector
3. Triangle
4. Write the translation rule for the vector translation from #3.
Answers:
1. Look at the movement from
2. The graph is the vector
3. It would be helpful to graph
4. To write
Practice
 What is the difference between a vector and a ray?
Use the translation
 What is the image of
A(−6,3) ?  What is the image of
B(4,8) ?  What is the preimage of
C′(5,−3) ?  What is the image of
A′ ?  What is the preimage of
D′(12,7) ?  What is the image of
A′′ ?  Plot
A,A′,A′′ , andA′′′ from the questions above. What do you notice? Write a conjecture.
The vertices of

(x,y)→(x−2,y−7) 
(x,y)→(x+11,y+4) 
(x,y)→(x,y−3) 
(x,y)→(x−5,y+8)
In questions 1316,
For questions 1719, name each vector and find its component form.
 The coordinates of
△DEF areD(4,−2),E(7,−4) andF(5,3) . Translate△DEF using the vector⟨5,11⟩ and find the coordinates of△D′E′F′ .  The coordinates of quadrilateral
QUAD areQ(−6,1),U(−3,7),A(4,−2) andD(1,−8) . TranslateQUAD using the vector⟨−3,−7⟩ and find the coordinates ofQ′U′A′D′ .
Center of Rotation
In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.Image
The image is the final appearance of a figure after a transformation operation.Preimage
The preimage is the original appearance of a figure in a transformation operation.Transformation
A transformation moves a figure in some way on the coordinate plane.Translation
A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.Image Attributions
Description
Learning Objectives
Here you'll learn what a translation is and how to perform translation rules.