Find a translation that has the same effect as the composition of translations below. $$ T_{(5,4)}(x, y) $$ followed by $$ T_{(=4,9)}(x, y) $$ Choose the correct answer below. A. $$ (x, y) \rightarrow(x+1, y=5) $$ B. $$ (x, y) \rightarrow(x+9, y-5) $$ C. $$ (x, y) \rightarrow(x+1, y+13) $$ D. $$ (x, y) \rightarrow(x+9, y+13) $$

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We start with the translation $$ T_{(5,4)} $$ which maps:
$$
(x, y) \rightarrow (x+5, y+4)
$$
Then, the translation $$ T_{(4,9)} $$ maps:
$$
(x, y) \rightarrow (x+4, y+9)
$$
When we compose these translations, we first apply $$ T_{(5,4)} $$ then $$ T_{(4,9)} $$. Thus, starting with $$(x, y)$$, we have:

1. After the first translation:
   $$
   (x, y) \rightarrow (x+5, y+4)
   $$

2. Apply the second translation to the result:
   $$
   (x+5, y+4) \rightarrow ((x+5)+4, (y+4)+9) = (x+9, y+13)
   $$

Therefore, the composition of the two translations is:
$$
(x, y) \rightarrow (x+9, y+13)
$$

The correct answer is option D.
The correct translation is $$ (x, y) \rightarrow (x+9, y+13) $$, which is option D.

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