The isosceles trapezoid shown has side lengths as labeled. How long is segment AC?

Draw  DE  perpendicular to  AB

Then  ADE  forms a right triangle  with  AE  =  (21 -9 ) / 2   = 12/2  =  6

Now....cos angle DAE  = (6/10)  =  3/5

And angle ABC  is supplemental to this...so its cos  =  -3/5

And we can use the Law of Cosines to find AC

AC  =  sqrt  [ AD^2 + DC^2  - 2(AD * DC)cos (ABC) ]

AC  = sqrt [ 10^2  + 9^2 - 2 (10 * 9) (-3/5) ]

AC  = sqrt [ 181  + 108 ]

AC = sqrt [289 ] =    17