What is the length of line segment g' a'? !

I believe the perimeter of the adult pinball machine is given as 172 in^2.
First, for the adult machine, we have:
GE = AM = 56 inch
EM = 35 inch
perimeter = 172 inch
Therefore:
GA = 172 - (56+56+35) = 172 - 147 = 25 inch

Now, the children version is a scaled down version of the adult version. This means that the sides of the version are proportional to their corresponding sides in the child version, therefore:
EM/E'M' = GA/G'A' = GE/G'E' = AM/A'M'
We have:
EM = 35 in
E'M' = 14 in
GA = 25 in
G'A' is the one we need to find

Substitute with the givens in the proportionality expression above to get G'A' as follows:
EM/E'M' = GA/G'A' 
35/14 = 25/G'A'
35 G'A' = 25*14
G'A' = (25*14) / 35
G'A' = 10 inch