Let's go through the solution step-by-step:
1. Identify the translation vector:
- The original coordinates for point [tex]\( C \)[/tex] are [tex]\( (2, 4) \)[/tex].
- The new coordinates for point [tex]\( C \)[/tex] after the translation are [tex]\( (3, 2) \)[/tex].
So, the translation vector [tex]\(\vec{T}\)[/tex] is calculated by:
[tex]\[
\vec{T} = (C' - C) = (3 - 2, 2 - 4) = (1, -2)
\][/tex]
2. Apply the translation vector to point [tex]\( A \)[/tex] (original coordinates [tex]\( (1, 1) \)[/tex]):
- Using the translation vector [tex]\(\vec{T}\)[/tex]:
[tex]\[
A' = A + \vec{T} = (1, 1) + (1, -2) = (1 + 1, 1 - 2) = (2, -1)
\][/tex]
3. Apply the translation vector to point [tex]\( R \)[/tex] (original coordinates [tex]\( (3, 0) \)[/tex]):
- Again, using the translation vector [tex]\(\vec{T}\)[/tex]:
[tex]\[
R' = R + \vec{T} = (3, 0) + (1, -2) = (3 + 1, 0 - 2) = (4, -2)
\][/tex]
4. Verification with given options:
- Comparing the new coordinates:
- For [tex]\( A' \)[/tex]: [tex]\( (2, -1) \)[/tex]
- For [tex]\( R' \)[/tex]: [tex]\( (4, -2) \)[/tex]
These coordinates match the third given option:
[tex]\[
A' = (2, -1) \quad \text{and} \quad R' = (4, -2).
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{3}
\][/tex]
