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We can use the Pythagorean theorem to find out the y relation to x, i.e. to express y in terms of x.

From the triangle PSM, PM2 = x2 + 92
From the triangle MRN, MN2 = (12-x)2 + (9-y)2
From the triangle PQN, PN2 = y2 + 122
From the triangle PMN, PN2 = PM2 + MN2

Thus, we have 4 equations that we can use it to replace the unknown in the last equation:

PN2 = PM2 + MN2

=> y2 + 122. = (x2 + 92 ) + (12-x)2 + (9-y)2
     
y2 + 122. = x2 + 92 + 122 - 24x + x2 + 92 -18y + y2
     0 = 2x2 + 2(9)2 -24x - 18y
  18y = 2x2 + 2(9)2 -24x 
  9y  = x2 + 92 - 12x  (this fulfills the first part of question, express y in terms of x) 

To find out the minimum value of y, we have to use the differentiation theory.
Let k = 9y, then  k = x2 + 92 - 12x 

d(k)/d(x) = 2(x)(2-1) - 1(12)x(1-0)
             = 2x - 12

For max or min values,  d(k)/d(x) = 0, 
so,    0 = 2x - 12
        2x = 12
         x =. 6

replace back x = 6 into the y expression above, 
9y  = x2 + 92 - 12x
9y  = 62 + 92 - 12(6)
     = 36 + 81 - 72
      = 45
 y   = 45/9 = 5 

the minimum value of y is 5 cm.

         

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