To determine the rule used for translating triangle [tex]\( EFG \)[/tex], we need to find the vector that translates each vertex of the original triangle to the corresponding vertex in the image. We will compare the original coordinates of [tex]\( E, F, \)[/tex] and [tex]\( G \)[/tex] with the translated coordinates [tex]\( E', F', \)[/tex] and [tex]\( G' \)[/tex].
### Step-by-Step Solution:
1. Find the translation for vertex [tex]\( E \)[/tex]:
- Original coordinates: [tex]\( E(-3, 4) \)[/tex]
- Translated coordinates: [tex]\( E'(-1, 0) \)[/tex]
Calculate the translation vector:
[tex]\[
\Delta x_E = -1 - (-3) = -1 + 3 = 2
\][/tex]
[tex]\[
\Delta y_E = 0 - 4 = -4
\][/tex]
Thus, the translation vector for [tex]\( E \)[/tex] is [tex]\((2, -4)\)[/tex].
2. Find the translation for vertex [tex]\( F \)[/tex]:
- Original coordinates: [tex]\( F(-5, -1) \)[/tex]
- Translated coordinates: [tex]\( F'(-3, -5) \)[/tex]
Calculate the translation vector:
[tex]\[
\Delta x_F = -3 - (-5) = -3 + 5 = 2
\][/tex]
[tex]\[
\Delta y_F = -5 - (-1) = -5 + 1 = -4
\][/tex]
Thus, the translation vector for [tex]\( F \)[/tex] is [tex]\((2, -4)\)[/tex].
3. Find the translation for vertex [tex]\( G \)[/tex]:
- Original coordinates: [tex]\( G(1, 1) \)[/tex]
- Translated coordinates: [tex]\( G'(3, -3) \)[/tex]
Calculate the translation vector:
[tex]\[
\Delta x_G = 3 - 1 = 2
\][/tex]
[tex]\[
\Delta y_G = -3 - 1 = -4
\][/tex]
Thus, the translation vector for [tex]\( G \)[/tex] is [tex]\((2, -4)\)[/tex].
4. Verify the consistency of the translation vector:
We have obtained the same translation vector [tex]\((2, -4)\)[/tex] for all three vertices [tex]\( E, F, \)[/tex] and [tex]\( G \)[/tex]. Therefore, the translation rule is:
[tex]\[
T_{2,-4}(x, y)
\][/tex]
Hence, the rule used to translate the image is [tex]\( \boxed{T_{2, -4}(x, y)} \)[/tex].
