geometry help (again)

Let's call the unlabeled side of the large triangle  " x ".

And let's call the height of that triangle  " h ". 

By Pythagorean theorem, we can make the equation

x² + y²   =   (16 + 4)²

                                    Subtract  y²  from both sides of the equation.

x²   =   (16 + 4)² - y²

Again by the Pythagorean theorem, we can make the equation

16² + h²   =   x²

                            Subtract  16²  from both sides of the equation.

h²   =   x² - 16²

                                             Substitute   (16 + 4)² - y²   in for   x²

h²   =   (16 + 4)² - y² - 16²

Again by the Pythagorean theorem, we can make the equation

4² + h²   =   y²               Substitute   (16 + 4)² - y² - 16²   in for  h²

4² + (16 + 4)² - y² - 16²   =   y²      Now we can solve for  y .

                                                     Simplify the left side of the equation.

4² + 20² - y² - 16²   =   y²

16 + 400 - y² - 256   =   y²

160 - y²  =  y²

                          Add  y²  to both sides of the equation.

160   =   2y²

                          Divide both sides by  2

80   =   y²

                          Take the positive square root of both sides.

√80  =  y

y  =  √[2*2 * 2*2 * 5]

y  =  4√5

Call the height, h

Since the apex angle  is a right angle... the height is forms a geometric mean, thusly 

16/h  = h/4      cross-multiply

h^2  =  64

h  =  8

And by the Pythagoren Theorem

√ [ h^2  + 4^2  ]  =  y

√ [ 8^2 + 4^2 ] = y

√80  =  y

√[16 * 5]   y

4√5  = y