Well the tooth might have fractured resulting in exposure of nerves leading to sensitivity to hot and cold. You need to visit dentist as early as possible to save the tooth either with root canal treatment if nerves are exposed or with the help of filling if not exposed which will be evident on an x-ray. Till that time you can have a pain killer to subside pain.

A tooth that breaks in that fashion generally in indicative of a problem either in the tooth itself or your bite. It can be restored with a bonded filling or a ceramic onlay. Please get it examined.

First of all , breaking of any tooth is not that easy the way you have explained, and it's not natural too. So fistand foremost is examining your complete jaw for the status of your rest of teeh, checking for any habits such as bittle nut chewing etc. Then we can plan a foolproof treatment for you. As of now , the broken part may be re-build by either composite filling or silver filling. In case if it's non-restorable one may plan for a crown (cap) on that molar. In worst situation if the fracture is deep (though it's not painful it's possible that it may be deep) one may advice root canal treatment and then putting a crown on that molar. GET IT EXAMINED WITH X-RAY AND RESTORE IT AT THE EARLIEST.

First of all please get your tooth examined by a professional dentist. According to the symptoms you said the tooth has broken because of decay which might be much deeper. If the situation is so then you will need a root canal treatment for that tooth. A thorough examination along with X-ray will give you the right diagnosis.