The answer is √2

We can use the Pythagorean Theorem.

We can solve for the hypotenuse of BPC: 1^2 + 1^2 = 2 -> √2 is the length of BC. We can solve for x, a leg of the triangle ABC with the pythagorean theorem again. 1 + 1 = 2 -> 2 + x^2 = 4 -> 4 - 2 = x^2 -> x^2 = 2 -> x = √2