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 } } \parallel \overline { b b ^ { \prime } } \parallel \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
other.
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, the segments connecting corresponding points (such as \( \overline{AA'} \), \( \overline{BB'} \), \( \overline{CC'} \)) represent the paths that each point of the pre - image takes to reach the image. The symbol \( \parallel \) means parallel. So, \( \overline{AA'} \parallel \overline{BB'} \parallel \overline{CC'} \) indicates that these paths (the segments connecting pre - image and image points) are parallel to each other.

• The first option is incorrect because \( \overline{AA'} \), \( \overline{BB'} \), \( \overline{CC'} \) are not the sides of the polygon \( \triangle ABC \) or \( \triangle A'B'C' \).
• The second option is incorrect as it misidentifies the segments (they are not sides of the polygon in the context of the translation property shown by the parallel symbol).
• The fourth option is about congruence (\( \cong \)), but the given symbol is \( \parallel \) (parallel), so it is incorrect.

Answer:

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