To determine the rule that describes the transformation from points [tex]\(E\)[/tex], [tex]\(F\)[/tex], and [tex]\(G\)[/tex] to their corresponding prime points [tex]\(E'\)[/tex], [tex]\(F'\)[/tex], and [tex]\(G'\)[/tex], we need to find a pattern or rule. We will accomplish this by analyzing how each point's coordinates change.
Let's analyze each pair of original and transformed points:
1. Point [tex]\(E\)[/tex] to [tex]\(E'\)[/tex]:
- Original point [tex]\(E = (-5, -3)\)[/tex]
- Transformed point [tex]\(E' = (-2.5, -1.5)\)[/tex]
To find the translation rule, we calculate how much each coordinate changes:
[tex]\[
x\text{-coordinate: } -2.5 - (-5) = -2.5 + 5 = 2.5
\][/tex]
[tex]\[
y\text{-coordinate: } -1.5 - (-3) = -1.5 + 3 = 1.5
\][/tex]
Thus, the translation from [tex]\(E\)[/tex] to [tex]\(E'\)[/tex] is by [tex]\((2.5, 1.5)\)[/tex].
2. Point [tex]\(F\)[/tex] to [tex]\(F'\)[/tex]:
- Original point [tex]\(F = (-5, 1)\)[/tex]
- Transformed point [tex]\(F' = (-2.5, 0.5)\)[/tex]
Again, we calculate the change in coordinates:
[tex]\[
x\text{-coordinate: } -2.5 - (-5) = -2.5 + 5 = 2.5
\][/tex]
[tex]\[
y\text{-coordinate: } 0.5 - 1 = -0.5
\][/tex]
Thus, the translation from [tex]\(F\)[/tex] to [tex]\(F'\)[/tex] is by [tex]\((2.5, -0.5)\)[/tex].
3. Point [tex]\(G\)[/tex] to [tex]\(G'\)[/tex]:
- Original point [tex]\(G = (-1, -2)\)[/tex]
- Transformed point [tex]\(G' = (-0.5, -1)\)[/tex]
Finally, we calculate the changes:
[tex]\[
x\text{-coordinate: } -0.5 - (-1) = -0.5 + 1 = 0.5
\][/tex]
[tex]\[
y\text{-coordinate: } -1 - (-2) = -1 + 2 = 1
\][/tex]
Thus, the translation from [tex]\(G\)[/tex] to [tex]\(G'\)[/tex] is by [tex]\((0.5, 1)\)[/tex].
Since we now have calculated the translation for each point, we observe that there are inconsistencies in translation magnitudes:
- [tex]\(E \to E'\)[/tex]: translation by [tex]\((2.5, 1.5)\)[/tex]
- [tex]\(F \to F'\)[/tex]: translation by [tex]\((2.5, -0.5)\)[/tex]
- [tex]\(G \to G'\)[/tex]: translation by [tex]\((0.5, 1)\)[/tex]
Since the translations are not consistent, the transformation cannot be solely a translation. Essentially, each point follows a different rule in coordinate transformation.
Therefore, the described transformation cannot be represented by a single uniform rule for translation. A more complex transformation or a combination of transformations might be involved, such as a combination of translation with other transformations (like scaling, rotation, or shearing). However, with the given details, specifying a single rule to broadly apply is impossible due to inconsistencies.
