YOU SAID:

The cat that eat hat did that

INTO JAPANESE

帽子を食べる猫はそれをしました

BACK INTO ENGLISH

the hat eating cat did it

INTO JAPANESE

帽子を食べる猫がやった

BACK INTO ENGLISH

the cat eating the hat did it

INTO JAPANESE

帽子を食べた猫がやった

BACK INTO ENGLISH

the cat that ate the hat did it

INTO JAPANESE

帽子を食べた猫がやった

BACK INTO ENGLISH

the cat that ate the hat did it