A tad excessive there, Skywalker.

I'm not saying Luke shouldn't have reacted when he found out Darth Vader was his father.
I think everybody can agree, that's a lot to take in.
But suicidally plunging into an abyss?
Maybe a bit dramatic.
I'm just saying, the guy is offering to rule the galaxy and betray his boss, Luke's enemy, the emperor.
He gets a dad. He squashes the empire. Probably some sick government insurance to get himself a better hand. If you really think about the moment, it could've gone really well for him.
I can understand though. He just flew in. Jet leg. Tensions were high. He wasn't all that sure about his relationship with Leia.  
Still though...
I'm just saying, maybe he could've gotten a DNA test before hurling himself to a highly probable demise. They were in a city in the clouds. They probably had fifteen gadgets and gizmos in that very room that could've tested their DNA. All Luke had to do was climb off the ledge and say, "Let's find out." Then walk up to the nearest droid and wait for it to boot up and say, "Please insert your right hand."
...Or maybe one calibrated for left hands...