To determine the rule used to translate the image of triangle [tex]\( EFG \)[/tex] to triangle [tex]\( E'F'G' \)[/tex], we need to identify the translation vector that maps the original vertices to their corresponding image vertices.
Let's determine the translation vector for each vertex:
1. For vertex [tex]\( E \)[/tex]:
- Original coordinates of [tex]\( E \)[/tex]: [tex]\( (-3, 4) \)[/tex]
- New coordinates of [tex]\( E' \)[/tex]: [tex]\( (-1, 0) \)[/tex]
- Translation vector: [tex]\( (E'_x - E_x, E'_y - E_y) = (-1 - (-3), 0 - 4) = (-1 + 3, 0 - 4) = (2, -4) \)[/tex]
2. For vertex [tex]\( F \)[/tex]:
- Original coordinates of [tex]\( F \)[/tex]: [tex]\( (-5, -1) \)[/tex]
- New coordinates of [tex]\( F' \)[/tex]: [tex]\( (-3, -5) \)[/tex]
- Translation vector: [tex]\( (F'_x - F_x, F'_y - F_y) = (-3 - (-5), -5 - (-1)) = (-3 + 5, -5 + 1) = (2, -4) \)[/tex]
3. For vertex [tex]\( G \)[/tex]:
- Original coordinates of [tex]\( G \)[/tex]: [tex]\( (1, 1) \)[/tex]
- New coordinates of [tex]\( G' \)[/tex]: [tex]\( (3, -3) \)[/tex]
- Translation vector: [tex]\( (G'_x - G_x, G'_y - G_y) = (3 - 1, -3 - 1) = (2, -4) \)[/tex]
All translations vectors are the same: [tex]\( (2, -4) \)[/tex].
Hence, the translation rule used to translate the triangle [tex]\( EFG \)[/tex] to [tex]\( E'F'G' \)[/tex] is [tex]\( T_{2, -4}(x, y) \)[/tex].
The correct answer is:
[tex]\[ T_{2, -4}(x, y) \][/tex]