Question

concept translations
a translation is a transformation in a plane
that maps all points of a preimage the same
distance and in the same direction.
the translation of \\( \triangle a b c \\) by \\( x \\) units along the
\\( x \\)-axis and by \\( y \\) units along the \\( y \\)-axis can be
written as \\( t _ { ( x, y ) } ( \triangle a b c ) = \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\).
a translation has the following properties:
if \\( t _ { ( x, y ) } ( \triangle a b c ) = \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\), then

• \\( \overline { a a ^ { \prime } } \parallel \overline { b b ^ { \prime } } \parallel \overline { c c ^ { \prime } } \\).
• \\( \overline { a a ^ { \prime } } \cong \overline { b b ^ { \prime } } \cong \overline { c c ^ { \prime } } \\).
• \\( \triangle a b c \\) and \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\) have the same orientation.

a translation is a rigid motion, so length and angle measure are preserved.

1. what does the mathematical

statement,
\\( \overline { a a ^ { \prime } } \cong \overline { b b ^ { \prime } } \cong \overline { c c ^ { \prime } } \\)
say for a translation?
the sides of a polygon of a
translation are all congruent to
each other.
the sides of a polygon of a
translation are all parallel to each
ot her.
the paths for each point in a
translation of a polygon are all
parallel to each other.
the paths for each point in a
translation of a polygon are all
congruent to each other.

Explanation:

In a translation, \( \overline{AA'} \), \( \overline{BB'} \), and \( \overline{CC'} \) represent the paths of points \( A \), \( B \), and \( C \) respectively. The congruence \( \overline{AA'} \cong \overline{BB'} \cong \overline{CC'} \) means these paths (segments) have the same length. Congruent segments have equal measures.

Answer:

The paths for each point in a translation of a polygon are all congruent to each other.