Greek - did persues kill medusa?

the story where perseus was the son of poseidon is from the book percy jackson and the olympians

Acrisius was locking Danae in a tower until Zues came. they had a baby. who was named Perseus. Acrisius got mad and trapped Danae and Perseus in a small chest and sent them out to sea. they arrived on an island where the fisherman was married. everyone bought a present but Perseus. so the fisherman asked for the head of Medusa. with Hermes and Athenas help he slayed Medusa while she slept. the fisherman married perseus mom while he was gone. he said "let all who are my friends shield their eyes." the head of medusa still worked so he raised the head and they turned to stone. eventully he was killed to Dionysus so he and his bride were turned into stars so they could live forever.