latin roots

I got in a word argument with my mom today because we were looking at the New York Times spelling bee words from yesterday and one of the words was 'arable' and for whatever reason I was very confident that it was related somehow to the word 'arid' and I believe my exact quote was something like

"look up 'arid' and tell me what it means and see if I'm right" said in a very demanding tone.

And I'll say right now that I was technically wrong but I did uncover this! Stay with me now.

Arable means used or suitable for growing crops and comes from the Latin word 'arare' meaning to plow.

Arid means too dry or barren to support vegetation and comes from the Latin word 'arere' meaning be dry or parched.

So, listen, 'arare' and 'arere' and they are related to ways to describe how suitable land is. That's not nothing. You can't tell me that's not nothing. Changing an 'a' to an 'e'. I think I've stumbled onto something much deeper here. 

Probably, probably some linguists in the comments can back me up on this that if we go back to the Proto-Indo-European roots we might find a point of connection. 

OH MY GOD I JUST LOOKED IT UP I WAS RIGHT!!

I'll give you the links and everything.

Look at the etymology part for arere: https://en.wiktionary.org/wiki/arere

I'll tell you where it goes you don't even have to look it goes to this: https://en.wiktionary.org/wiki/aratrum#Latin

Just like I said. Just like I said the Proto-Indo-European word for plough. Same root. They're the same word. I was right. 

This is what my education was FOR. What do you think about that, Mom?!