Ram upgrade

MacTracker shows that the iMac (27-inch, Late 2009) and iMac (27-inch, Mid-2010) both had that model number. If you select About This Mac in the Apple () menu, you can further narrow this down. You should do this, as the two models had different requirements for the speed of their DDR3 SDRAM.

These iMacs have four slots for RAM modules. The official limit is 16 GB of RAM. The actual limit is 32 GB.

Install memory in an iMac - Apple Support

Other World Computing sells RAM for these machines.

https://eshop.macsales.com/shop/apple/memory/iMac

Crucial does not carry RAM for the Late 2009 model any more. They do carry RAM for the mid-2010 one. Their compatibility matching tool has two entries for 27-inch Mid-2010 systems, depending on whether your CPU is a Core i3 or not.

https://www.crucial.com/upgrades

There may be other suppliers, but you do want to correctly identify your specific iMac model, and to make sure that the RAM modules that you install meet Apple's specifications (see linked support document).

Another useful piece of information to have is the Model Identifier (e.g., "iMac11,1") from System Report.

Select About This Mac in the Apple () menu, then click on the System Report… button.