Lines in a rectangle with a specific point $P$

Question

I came across this interesting problem that might teach me something new.

On the above diagram, $ABCD$ is a rectangle and point $P$ such that $PB = 4\sqrt{2}$, $PC = 4$ and $PD= 3$.

Our goal is to find $PA$. If this was a problem that has good description I would just search up it, however, this is not easy to describe to a search engine, so here's my actual question: How do you find a length of a given line that is in a rectangle connected to some point inside the rectangle and knowing how long each of the other lines connecting the rectangle's corner with the specific point? (Hopefully you didn't have a headache reading this)

I suspect there is some sort of formula to figure this out, so any help would be greatly appreciated!

Answer 1

Hint

Draw parallel lines from point $P$ to the all four sides of the rectangle. Apply the Pythagorean Theorem four times and solve the system of equations.

You should obtain: $PB^2+PD^2=PA^2+PC^2$

Answer 2

Let's draw 4 perpendecular lines on all four edges from point P. The lines meet the edges at points L, R, U, D. I will call the heights: $h_{L}$, $h_{R}$, $h_{U}$, $h_{D}$

$$ PB^2 - PL^2 = LB^2 = CR^2 = CP^2 - PR^2 $$ Thus: $$ 32 - 16 = PL^2 - PR^2 $$

Similarly:

$$ PA^2 - PD^2 = ...= PL^2 - PR^2 $$

Thus, according to the two previous equations: $$ PA^2 - 9 = 16 $$ $$PA = 5$$