The short answer is the cell is eukaryotic because it has chloroplasts.
The long answer is eukaryotic cells have the ability to partition off different functions to locations within the cell, think compartments. These are called organelles and they evolved for this purpose. Chloroplasts are examples of organelles and are only found in eukaryotic cells. Eukaryotic cells have a nucleus with their DNA; where prokaryotic do not have a nucleus. Both prokaryotes and eukaryotes possess a cell membrane and cytoplasm. Remember that prokaryotic cells do not contain a nucleus or any other membrane-bound organelle.
According to South Dakota Department of Health, these are some key differences between the two:
- Usually single celled organisms
- Contain no nucleus
- No membrane bound cell structures
- Single or multi-celled
- Have a nucleus
- Contain a nuclear envelope
- Contain cell membrane bound structures like mitochondria, golgi apparatus,etc.