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Name
2.1
Class
Date
Translations
Essential Question: How do you draw the image of a figure under a translation?
Resource
Locker
Exploring Translations
Explore
A translation slides all points of a figure the same distance in the same direction.
You can use tracing paper to model translating a triangle.
A
First, draw a triangle on lined paper. Label the
vertices A, B, and C. Then draw a line segment
XY. An example of what your drawing may look
like is shown.
B
Use tracing paper to draw a copy of triangle
_
ABC. Then copy XY so that the point X is on top
of point A. Label the point made from Y as A'.
B
B
A
A
C
A
Y
X
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C
C
Using the same piece of tracing paper, place
A' on A and draw a copy of △ABC. Label the
corresponding vertices B' and C'. An example
of what your drawing may look like is shown.
D
Use a ruler to draw line segments from each
vertex of the preimage to the corresponding
vertex on the new image.
B
B
B
B
A
A
A
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C'
C
X
Y
X
A
Y
X
63
C'
C
Y
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E
DO NOT
Correcti
Measure the distances AA', BB', CC', and XY. Describe how AA', BB', and CC' compare to
the length XY.
Possible answer: BB', AA', and CC' are each 2 inches. XY is also 2 inches. The segments are
all the same length.
Reflect
1.
Are BB', AA', and CC' parallel, perpendicular, or neither? Describe how you can check that your
answer is reasonable.
They are parallel. I can turn the tracing paper so that one of the lines is on one of the
parallel rules of the lined paper. All of the segments line up with the lines on the paper.
So, they are parallel.
2.
How does the angle BAC relate to the angle B'A'C' ? Explain.
Possible answer: The angles are congruent. Since translation is a rigid transformation, the
side lengths and angle measures remain the same in the translated figure.
Explain 1
Translating Figures Using Vectors
A vector is a quantity that has both direction and magnitude.
The initial point of a vector is the starting point.
The terminal point of a vector is the ending point. The vector
⇀
shown may be named ‾ EF or ⇀
v.
F
⇀
ν
E
Initial
point
Terminal
point
Translation
It is convenient to describe translations using vectors. A translation is a transformation along
a vector such that the segment joining a point and its image has the same length as the vector
and is parallel to the vector.
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A′
A
C′
B
C
B′
⇀
ν
For example, BB' is a line segment that is the same length as and is parallel to vector ⇀
v.
You can use these facts about parallel lines to draw translations.
• Parallel lines are always the same distance apart and never intersect.
• Parallel lines have the same slope.
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Example 1 Draw the image of
△ABC after a translation along ⇀
v.

A
B
⇀
ν
C
Draw a copy of⇀
v with its initial point at
vertex A of △ABC. The copy must be
the same length as ⇀
v , and it must be
parallel to ⇀
v . Repeat this process at
vertices B and C.
Draw segments to connect the
terminal points of the vectors.
Label the points A', B', and C'.
△A'B'C' is the image of △ABC.
A′
A
A
B′
B
B
C′
⇀
ν
C
A′

B′
A
D
⇀
ν
Draw a vector from the vertexA that is the same length as
_
and parallel to vector v. The terminal point A' will
left
5
be
units up and 3 units
.
C′
D′
⇀
ν
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C
Draw three more vectors that are parallel from,
and D with terminal points B', C', and D'.
B
C
B
,
C
,
Draw segments connecting A', B', C', and D' to
form quadrilateral A'B'C'D' .
Reflect
3.
How is drawing an image of quadrilateral ABCD like drawing an image of
△ABC? How is it different?
Possible answer: The steps to drawing an image of the quadrilateral are the same as
drawing an image of the triangle. It is different because there is an extra vector when
drawing the quadrilateral. Also, you must be careful to connect the vertices in the same
order as the original shape.
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A
Your Turn
4.
Draw the image of △ABC after a translation along ⇀
v.
B
A′
B′
C
C′
⇀
ν
Drawing Translations on a Coordinate Plane
Explain 2
A vector can also be named using component form, 〈a, b〉, which specifies
the horizontal change a and the vertical change
_b from the initial point to
the terminal point. The component form for PQ is 〈5, 3〉.
Q
3
You can use the component form of the vector to draw coordinates for a new image on a
coordinate plane. By using this vector to move a shape, you are moving thex-coordinate
5 units to the right. So, the new x-coordinate would be 5 greater than the x-coordinate in
the preimage. Using this vector you are also moving the y-coordinate up 3 units. So, the
new y-coordinate would be 3 greater than the y-coordinate in the preimage.
P
5
Rules for Translations on a Coordinate Plane
Translation a units to the right
Translation a units to the left
Translation b units up
Translation b units down
(x, y) → (x + a, y)
(x, y) → (x – a, y)
(x, y) → (x, y + b)
(x, y) → (x, y – b)
So, when you move an image to the righta units and up b units, you use the rule
(x, y) → (x + a, y + b) which is the same as moving the image along vector 〈a, b〉.
 Preimage coordinates:
(−2, 1), (−3, -2) , and(−1, −2).Vector: 〈4, 6〉
Predict which quadrant the new image will be
drawn in: 1stquadrant.
Then use the preimage coordinates to
draw the preimage, and use the image
coordinates to draw the new image.
Use a table to record the new coordinates.
Use vector components to write the
transformation rule.
Preimage
coordinates
(x, y)
Image
(x + 4, y + 6)
(–3, –2)
(1,4)
(–2, 1)
(–1, –2)
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© Houghton Mifflin Harcourt Publishing Company
Example 2 Calculate the vertices of the image figure. Graph the preimage and the image.
y
A
6
4
A′
(2,7)
-2
C′
(3,4)
66
2
0
C
B
⇀
ν
2
x
4
B′
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A(3, 0), B(2, −2),and C(4, −2).Vector 〈−2, 3〉
 Preimage coordinates:
1
Prediction: The image will be in Quadrant
(
Preimage coordinates
(x, y)
.
)
Image
x− 2 ,y+ 3
(3, 0)
(2, −2)
(4, −2)
(1
(0
(2
,
,
,
)
1 )
1 )
y
4
A′
2
3
-2
C′
B′
0
x
A
2
-2
B
4
C
Your Turn
Draw the preimage and image of each triangle under a translation along 〈−4, 1〉.
5.
Triangle with coordinates:
A(2, 4), B(1, 2), C(4, 2).
4
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y
A
C′
2
-2
Triangle with coordinates:
P(2, –1), Q(2, –3), R(4, –3).
y
A′
B′
6.
B
2
C
0
2
P′
-2
x
Q′
4
x
0
-2
2P
R′
-2
Q
4
R
Specifying Translation Vectors
Explain 3
You may be asked to specify a translation that carries a given figure onto another figure.
You can do this by drawing the translation vector and then writing it in component form.
Example 3 Specify the component form of the vector that maps

A
4
C
Determine the components of ⇀
v.
y
2
B
-4
⇀
ν
0
- 4 B′
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x
A′
-2
△ABC to △A'B'C'.
2
4
C′
The horizontal change from the initial point −
( 4, 1) to the terminal
point( 1, −3)is 1 − (−4) = 5.
The vertical change from the initial point −4,
( 1) to the terminal point
(1, −3)is −3 − 1 = −4
Write the vector in component form.
⇀
v = 〈5, -4〉
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Draw the vector ⇀
v from a vertex of △ABC to its image
in △A'B'C'.
y
B
4
⇀
ν
A
C
-2
2
B
0
-2
A′
B′
Determine the components of ⇀
v.
C′
2
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x
4
The horizontal change from the initial point (–3, 1) to the terminal
point (2, 4) is 2 – -3 = 5 .
The vertical change from the initial point to the terminal point
is 4 – 1 = 3
Write the vector in component form. ⇀
v =
⟨
5 , 3
⟩
Reflect
7.
What is the component form of a vector that translates figures horizontally? Explain.
The vector has the form ⟨a, 0⟩. There is no change in the vertical direction, so the value of
b is 0.
Your Turn
8.
In Example 3A, suppose △A'B'C' is the preimage and △ABC is the image after
translation. What is the component form of the translation vector in this case?
How is this vector related to the vector you wrote in Example 3A?
〈-5, 4〉; the components are the opposites of the components of the vector in Example 3A.
Elaborate
9.
How are translations along the vectors 〈a, −b〉 and 〈−a, b〉 similar and how are they different?
They move the points the same distance, but in opposite directions.
11. A translation along the vector 〈a, b〉 maps points in Quadrant I to points in
Quadrant III. What can you conclude about a and b? Justify your response.
Both a and b are negative. Points in Quadrant I have positive x- and y-coordinates and
points in Quadrant III have negative x- and y-coordinates. This means the horizontal and
vertical changes are both negative. So a is negative and b is negative.
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10. A translation along the vector 〈−2, 7〉 maps point P to point Q. The coordinates of
point Q are (4, −1). What are the coordinates of point P? Explain your reasoning.
(6, −8); Solving equations x − 2 = 4 and y + 7 = −1 shows that x = 6 and y = −8.
12. Essential Question Check-In How does translating a figure using the formal
definition of a translation compare to the previous method of translating a figure?
Possible answer: Rather than sliding a figure left or right and then up or down to translate
it, you slide it parallel to a given vector a distance equal to the length of the vector.
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