Final answer:
Icarus and Daedalus were imprisoned on the island by King Minos of Crete because Daedalus was a threat to Minos due to his knowledge of the Labyrinth which he designed and his involvement in betrayals against Minos.
Step-by-step explanation:
The reason Icarus and Daedalus are imprisoned on the island is associated with the Greek mythological story that tells of Daedalus's ingenuity and Icarus's tragic downfall. Daedalus, a skilled craftsman, created the Labyrinth for King Minos of Crete to imprison the Minotaur.
However, due to various reasons including Daedalus's own betrayal of Minos by helping Minos's wife and others navigate the Labyrinth, Daedalus and his son Icarus were imprisoned by Minos to prevent Daedalus from spreading the knowledge of the Labyrinth.
Their escape from imprisonment led to the crafting of wings made of feathers and wax. The story culminates in Icarus's fall when, in his hubris, he flies too close to the sun, melting his wax wings, leading to his death while Daedalus manages to escape to safety.