Step 1: In a square, the diagonals bisect each other. So $P$ is the midpoint of $BD$.
Step 2: Midpoint formula :
$$P = \left(\frac{9+1}{2},\ \frac{-2+6}{2}\right) = \left(\frac{10}{2},\ \frac{4}{2}\right) = (5,\ 2)$$
Hence $P(5, 2)$.

Step 3: Length of diagonal $BD$ by distance formula :
$$BD = \sqrt{(1-9)^2 + (6-(-2))^2} = \sqrt{(-8)^2 + (8)^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}$$
Step 4: For a square of side $a$, diagonal $= a\sqrt{2}$.
$$a\sqrt{2} = 8\sqrt{2} \Rightarrow a = 8$$
Hence the side of the square $= 8$ units.