right triangle

x+y = sqrt(92 + 122) = 15

z = (9 * 12) / 15  = 7.2

x+y+z = 22.2 units

We  have a   9 - 12  - 15  right triangle

So  x +  y =  15  =  base length of this triangle

The  area of this triangle  =(1/2) (9)(12)  =   (1/2) ( 108)  =  54

The area can also be expressed as

54   =  (1/2) base * z

54  = (1/2) 15 * z

108 = 15 * z

z = 108/15 =  36/5

So

x + y + z  =    15  + 36/5 =     [ 75 + 36 ]  /  5  =   111  / 5

Because this triangle is a right triangle, by the Pythagorean theorem, we can figure out the hypotenuse by square rooting the sum of the square of the two legs, 9^2+12^2=sqrt225. The sqrt of 225 is 15 so x+y=15. Now we figure out the area of the big triangle. 9*12/2=54. Another way of calculating the area is by multiplying 15 by z and dividing by 2. Since we already figured out the area of the big triangle is 54, we can substitute 54 as the area and we get 54=z*15/2. Solving for z we get 7.2. x+y+z=15+7.2=22.2