How to Find the value of Z, Using the X and Y, between a three point plane

Question

I would like to know how to work out the value $z$, using $x$ and $y$ values, between 3 points.

For example if point A is (2,4,0), B is (10,3,3) and C is (2,10,1) D is (6,5,?) the ? being the unknown $z$ value i want to find

Thank you

Answer 1

The equation of a plane is $ax+by+cz=1$.

We have $2a+4b=1$, $10a+3b+3c=1$, and $2a+10b+c=1$.

Solving gives us $a=\frac{19}{70}, b=\frac{8}{70}, c=-\frac{48}{70}$.

Plugging this into $ax+by+cz=1$ and (x,y)=(6,5) yields z=$\boxed{\frac{7}{4}}$.