망가뜨리다 = 망가트리다 => break or ruin something beyond repair.
-뜨리다 and -트리다 are endings meaning "cause (something bad)" that appear in some verbs.
They are one and the same word except for the sightly different pronunciation (both are accepted as equally correct).
A related word is 망가지다, which means break(intransitive).
So you can say:
- 아이가 장난감을 망가뜨렸다: The child has broken his/her toy.
- 아이의 손에 장난감이 망가졌다: The toy has been broken by the child.