To determine the rule that describes the translation of a triangle on the coordinate plane, we need to carefully examine the given transformation.
### Problem Statement:
A triangle is translated:
1. 4 units to the right.
2. 3 units down.
### Translation Break Down:
1. Translating 4 units to the right:
- Translating a point to the right increases its [tex]\( x \)[/tex]-coordinate.
- This can be represented as [tex]\( x \rightarrow x + 4 \)[/tex].
2. Translating 3 units down:
- Translating a point downward decreases its [tex]\( y \)[/tex]-coordinate.
- This can be represented as [tex]\( y \rightarrow y - 3 \)[/tex].
Combining these two changes, the translation rule for any point [tex]\((x, y)\)[/tex] will be:
[tex]\[
(x, y) \rightarrow (x + 4, y - 3)
\][/tex]
### Matching with Options Provided:
- [tex]\((x, y) \rightarrow (x+3, y-4)\)[/tex]: This represents a translation of 3 units to the right and 4 units down, which does not match our required transformation.
- [tex]\((x, y) \rightarrow (x+3, y+4)\)[/tex]: This represents a translation of 3 units to the right and 4 units up, which also does not match.
- [tex]\((x, y) \rightarrow (x+4, y-3)\)[/tex]: This represents a translation of 4 units to the right and 3 units down, which matches our required transformation.
- [tex]\((x \rightarrow x+4, y \rightarrow y+3)\)[/tex]: This represents a translation of 4 units to the right and 3 units up, which does not match our required transformation.
### Conclusion:
The rule that correctly describes the required translation of moving 4 units to the right and 3 units down is:
[tex]\[
(x, y) \rightarrow (x + 4, y - 3)
\][/tex]
Among the given options, this corresponds to the third choice:
[tex]\[
\boxed{(x, y) \rightarrow (x+4, y-3)}
\][/tex]
Therefore, the correct answer is the third option.
