Chapter 11:
The Mathematics
of Symmetry
11.4 Translations
Bell Work
Try this one from the 11.3 worksheet we started
yesterday.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-2
Excursions in Modern Mathematics, 7e: 1.1 - 2
Translation
A translation consists of essentially dragging an object
in a specified direction and by a specified amount
(the length of the translation).
The two pieces of information (direction and length of
the translation) are combined in the form of a vector
of translation (usually denoted by v). The vector of
translation is represented by an arrow–the arrow
points in the direction of translation and the length of
the arrow is the length of the translation.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-3
Excursions in Modern Mathematics, 7e: 1.1 - 3
Translation
A very good illustration of a translation in a twodimensional plane is the dragging of the cursor on a
computer screen. Regardless of what happens in
between, the net result when you drag an icon on
your screen is a translation in a specific direction and
by a specific length.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-4
Excursions in Modern Mathematics, 7e: 1.1 - 4
Example 11.4
Translation of a Triangle
This figure illustrates the translation of a triangle ABC.
Two “different” arrows are shown in the figure, but they
both have the same length and direction, so they
describe the same vector of translation v.
As long as the arrow points in
the proper direction and has
the right length, the placement
of the arrow in the picture is
immaterial.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-5
Excursions in Modern Mathematics, 7e: 1.1 - 5
Properties of Translations
The following are some important properties of a
translation.
Property 1
If we are given a point P and its image P´ under a
translation, the arrow joining P to P´ gives the vector
of the translation. Once we know the vector of the
translation, we know where the translation moves any
other point. Thus, a single point-image pair P and P´ is
all we need to completely determine the translation.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-6
Excursions in Modern Mathematics, 7e: 1.1 - 6
Properties of Translations
Property 2
In a translation, every point gets moved some
distance and in some direction, so a translation has
no fixed points.
Property 3
When an object is translated, left-right and clockwisecounterclockwise orientations are preserved: A
translated left hand is still a left hand, and the hands
of a translated clock still move in the clock-wise
direction. In other words, translations are proper rigid
motions.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-7
Excursions in Modern Mathematics, 7e: 1.1 - 7
Properties of Translations
Property 4
The effect of a translation with vector v can be
undone by a translation of the same length but in the
opposite direction. The vector for this opposite
translation can be conveniently described as –v. Thus,
a translation with vector v followed with a translation
with vector –v is equivalent to the identity motion.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-8
Excursions in Modern Mathematics, 7e: 1.1 - 8
PROPERTIES OF TRANSLATIONS
■ A translation is completely determined by a
single point-image pair P and P´.
■ A translation has no fixed points.
■ A translation is a proper rigid motion.
■ When a translation with vector v is followed
with a translation with vector –v we get to the
identity motion.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-9
Excursions in Modern Mathematics, 7e: 1.1 - 9
Example
Find the image of the triangle under the translation if
A’ is shown.
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-10
Excursions in Modern Mathematics, 7e: 1.1 - 10
11.4 Practice
(pg. 348-349) #23-28
Copyright©
© 2014
2010 Pearson
Education,
Inc.
Copyright
Pearson
Education.
All rights reserved.
11.4-11
Excursions in Modern Mathematics, 7e: 1.1 - 11