To solve this problem, we need to apply the given translation to the coordinates of point [tex]\( C \)[/tex]. Let's go through the steps of this process:
1. Identify the original coordinates of point [tex]\( C \)[/tex]:
[tex]\[
C(-2, 3)
\][/tex]
2. Understand the translation vector, which states that each point [tex]\((x, y)\)[/tex] will be translated to [tex]\( (x-2, y-5) \)[/tex]:
[tex]\[
\text{Translation vector:} (-2, -5)
\][/tex]
3. Apply the translation to the coordinates of [tex]\( C \)[/tex]:
- The x-coordinate of [tex]\( C \)[/tex] is [tex]\(-2\)[/tex]. After applying the translation, this coordinate becomes:
[tex]\[
-2 + (-2) = -2 - 2 = -4
\][/tex]
- The y-coordinate of [tex]\( C \)[/tex] is [tex]\( 3 \)[/tex]. After applying the translation, this coordinate becomes:
[tex]\[
3 + (-5) = 3 - 5 = -2
\][/tex]
4. Combine the new x and y coordinates to find the translated coordinates of [tex]\( C \)[/tex]:
[tex]\[
C'(-4, -2)
\][/tex]
Thus, the coordinates for [tex]\( C' \)[/tex] after translation are [tex]\( (-4, -2) \)[/tex]. Therefore, the correct answer is:
[tex]\[ C^{\prime}(-4, -2) \][/tex]
