The radius is 10, so the diameter is 20. This means the parts of the diameter (the chord through the center P) to the left and right of the vertical chord have lengths 16 and 4, respectively.
Because the horizontal chord is a diameter, the vertical chord is cut in half, so its parts above and below the diameter both have length x.
Now, by the intersecting chord theorem,
16×4 = x × x
or
x ² = 64
so that
x = 8