Question 979444
The two diagonals intersect at point S.
(That must be what you meant when you wrote that the "middle letter of the parallelogram is S").
 
A property of parallelograms is that
"diagonals bisect each other",
meaning that they intersect at their midpoint.
So, {{{S}}} is the midpoint of diagonal {{{PN}}}, and the midpoint of diagonal {{{OM}}} .
Since S is the midpoint of {{{PN}}} , {{{PN=PS+SN}}} , and {{{PS=SN}}} ,
Then {{{system(PN=PS+SN,PS=SN)}}}--->{{{PN=PS+PS=2PS}}} , and also, since {{{PN=32cm}}} , {{{system(PN=2PS,PN=32cm)}}}--->{{{2PS=32cm}}}--->{{{PS=32cm/2}}}--->{{{highlight(PS=16cm)}}} .
Since S is the midpoint of {{{OM}}} , {{{OM=OS+SM}}} , and {{{OS=SM}}}
Then {{{system(OM=OS+SM,OS=SM)}}}--->{{{OM=OS+OS=2OS}}} , and also, since {{{OM=24cm}}} , {{{system(OM=2OS,OM=24cm)}}}--->{{{2OS=24cm}}}--->{{{OS=24cm/2}}}--->{{{highlight(OS=12cm)}}} .