Distances from a point inside a rectangle

\(P\) is a point inside rectangle \(ABCD\). The perpendiculars from \(P\) to the four sides have lengths \(w\), \(x\), \(y\) and \(z\), as shown in the diagram.

1. Prove that \(PA^2+PC^2=PB^2+PD^2\).[3]
2. Given that \(PA=3\) cm, \(PB=4\) cm and \(PC=5\) cm, find the exact value of \(PD\).[2]
3. A student claims there is a point inside some rectangle with \(PA=1\) cm, \(PB=7\) cm and \(PC=1\) cm. Use (a) to show that the claim is false.[2]