​Determine the value of $a$.

Connect   WY

Then triangle  WXY  is an isosceles  right triangle with leg WX  = leg XY

Using the  Pythagorean Theorem, we  can find  WY  as

√[  WX^2 + XY^2  ]   = √ [4^2 + 4^2 ] =  √32  =  4√2

And since  WX  = XY  then  angle WYX  =45°

So in triangle WYZ, angle  WYZ  = 135  -45  =  90°

So.....triangle  WYZ is also right WY and YZ  being legs and WZ the hypotenuse

So

√ [ WZ^2  - WY^2 ]  =  YZ

√[9^2 -32]  =  √[81 - 32]  = √49  = 7  =  YZ  =  "a"