The correct answer is 'polluted'.
Here's why:
* 'Badly' is an adverb that modifies a verb or adjective. In this case, it needs to modify the adjective that describes the state of the lake.
* 'Polluted' is the past participle of the verb 'to pollute', and it functions as an adjective here, describing the lake.
Therefore, the complete and correct sentence is: 'The lake is badly polluted.'
Correct answer: polluted