Let the perpendicular on to BC meet BC at M, say. Then BC = BM + MC.
Triangle AMC has one right-angle, and one of 45o, so the other angle is also 45o. This means that the sides AM and MC are of equal length. So length of MC = 8 .
Triangle AMB is also right-angled, with one side of length 8, and a hypotenuse of length 10. So the length of BM = √(100 - 64) = 6.
Hence BC = BM + MC = 6 + 8 = 14.
Use the pythagoreum Theorum
BC is twice the lenght of the side we just calculated.
So the answer is a....12.