We are tasked with translating the rectangle $ABCD$ horizontally by $-4$ and vertically by $+3$, to find the coordinates of $C'$, the translated position of $C$.

Step-by-Step Solution:

The current coordinates of $C$ are $(-5, 1)$. To perform the translation:

1. Horizontal Translation: Subtract 4 from the x-coordinate. $x' = -5 - 4 = -9$
2. Vertical Translation: Add 3 to the y-coordinate. $y' = 1 + 3 = 4$

Thus, the new coordinates of $C'$ after translation are: $C' = (-9, 4)$

Would you like a detailed explanation of translations or have further questions? 😊
Here are 5 related questions to deepen your understanding:

1. What are the new coordinates of $A'$, $B'$, and $D'$ after the same translation?
2. How does a translation differ from a reflection or rotation in geometry?
3. If the rectangle were translated vertically by $-2$ instead of $+3$, how would the coordinates of $C'$ change?
4. What is the mathematical rule for translating any point $(x, y)$ by $(h, k)$?
5. Can this problem be extended to three dimensions? If so, how?

Tip: Always double-check the direction of translations—horizontal (x-axis) and vertical (y-axis) affect the coordinates differently.