How would you go about cutting a part in half across a certain plane?

I was looking through snack break once again, trying to improve in scripting with lua. I came across a topic that looked easy until the last one. The topic was cutting a part relative to where another part was slicing through it. I assumed the first one would be easy, all I had to do was get the difference in position between the two points and then subtract that number from one half of the size of the part. All that was left was the positioning, and then I saw the third part of the challenge. The part was getting cut diagonally. This amazed me, I didn't think you could create a triangle part without negations, and I know you can't negate anything in a script. My question is, how could you cut a part across a diagonal plane with a script?

Note: I'm not expecting a full code answer, as that would take quite a while. I'm simply asking for an explanation on how this would work, and what API I could use to get this effect.