Determine the value of a.

Can't tell what the letter at the top left of the figure is...but...I think it is "H"

Anyway....join  H and Y 

Triangle HAY   is a isoceles right triangle.....with HX = XY

So....

HY =  sqrt [ HX^2 + XY^2]  =  sqrt [ 4^2 + 4^2 ] =  sqrt [ 32]   

And......HY will create  triangle HYZ

Angle HYX =  135 - Angle  HYX

Angle HYX = 135 - 45  = 90

So....triangle HYZ is right wth HZ the hypotenuse  and HY one of the legs

So

"a"  = ZY  =  sqrt [ HZ^2 - HY^2]  =  sqrt [ 9^2 - 32]  = sqrt [ 81 - 32] = sqrt  [49]  =  7