Chapter 10 Continued
10.2 Slides or Translations
• Translations
o Definition: A translation (or slide) determined by a ray l and a specified
distance is a translation that maps the point M onto point M’ so that the ray
MM’ is parallel to and in the same direction as l and MM’ equals the specified
distance.
o If each point P in a plane corresponds to a unique point in the plane, P’, such
that directed segments PP” is congruent and parallel to the directed segment
AB, then the correspondence is called the translation associated with the
directed segment AB and is written TAB
o TAB(P) = P’
A
C
D
A'
C'
D'
E
B
B
F
A
B'
TEF ABCD
• Translation on a coordinate system
o Definition: If P(x,y) is translated r units in an x direction and s units in a y
direction (denoted Tr, s), the image of P is P’(x + r, y + s).
4
A
B
2
D
C
-5
5
A'
B'
-2
D'
-4
C'
T(3,-4)
• Translation as a composite of reflections
o Definition: Let Rm and Rn represent two reflections about lines m and n,
respectively. Then the transformation “apply Rm and then apply Rn to the
image obtained by Rm” is called the composite of Rm and Rn and is denoted by
composite Rn(Rm).
o Definition: A transformation is a translation (T), or slide, if and only if there
exist parallel lines m and n such that T = composite Rn(Rm), where Rm is a
reflection over line m and Rn is a reflection over line n.
A
D
B
C
n
m
Original ABCD
A
A'
D
D'
B
B'
C
C'
n
m
Rm ABCD
A''
A
D''
A'
D
D'
B''
B
B'
C''
C
C'
n
m
Rn (Rm) ABCD
A''
A
D''
A'
D
D'
B''
B
B'
C''
C
C'
n
m

T
AA ''
o Translations and parallel lines: Definition: If a translation maps A onto A”,
then the line segment AA” is perpendicular to each of a set of parallel lines
when the translation is considered in terms of reflections and AA” is twice the
distance between the parallel lines.
o Parallel image: Definition: Under a translation of a line in which the direction
of the translation is not parallel to the line, the line and its image are parallel.
B''
B'
A''
P
n
A'
B
O
A
m
AB  A '' B ''
• Exercise Set 2 (p. 492)
o Do problems: 1-27 odd