Finder quit unexpectantly error message

Please determine if the same problems occur in "Safe Mode": Start up your Mac in safe mode - Apple Support.

For your particular model iMac, starting it in "Safe Mode" is accomplished by pressing and holding a Shift key while you use another finger to tap the power button in back. It works with either wired or wireless keyboards. Those instructions are described in that document, under If using any other Mac.

The lack of original system installation media complicates things, but there are workarounds depending on the exact model iMac (approximately 2010 is not exact enough) as well as how much time, money and effort you wish to devote to the task.

You can try booting OS X Recovery: How to start up from macOS Recovery - Apple Support

Follow the instructions for Intel-based Mac. If the Mac is too old to boot Recovery it will just ignore the attempt.

Thanks, that is considered a "Late 2009" model iMac, and its model identifier is iMac10,1

Do you have a Time Machine backup? Restoring a backup would be ideal.

If you have no backup to restore, that Mac is too old to use OS X Internet Recovery, and you will require those discs. Another alternative would be to purchase a retail Mac OS X Snow Leopard Install DVD, which Apple has not sold in many years. They shipped in a cardboard box and are white in color, and you can find them on a popular auction site. Caveat Emptor as always. Even if you find one and can install Snow Leopard on that Mac, you will probably need to erase it first.

Time, money and effort, that's what it will take.

John Galt wrote:

If you have no backup to restore, that Mac is too old to use OS X Internet Recovery,

FYI, while that model did not ship from the factory with Internet Recovery Mode access.......if macOS 10.13 was installed on it (the only 2009 iMac which was compatible with High Sierra), then it most likely does have access to Internet Recovery Mode from the firmware update included with that upgrade. It certainly doesn't hurt to try accessing Internet Recovery Mode using Command + Option + R.

No it won't hurt to try, but Computers that can be upgraded to use OS X Internet Recovery - Apple Support includes the following passage, followed by a list of the eligible Macs:

Download and install an updated EFI Firmware ROM for these computers to use the OS X Internet Recovery feature:

... and the last time I tried it on mine (upgraded to HS) it didn't work.

It did work for the mid-2010 and 2011 models for which I happen to recall installing that EFI Firmware upgrade.

John Galt wrote:

No it won't hurt to try, but Computers that can be upgraded to use OS X Internet Recovery - Apple Support includes the following passage, followed by a list of the eligible Macs:

Download and install an updated EFI Firmware ROM for these computers to use the OS X Internet Recovery feature:

... and the last time I tried it on mine (upgraded to HS) it didn't work.

It did work for the mid-2010 and 2011 models for which I happen to recall installing that EFI Firmware upgrade.

Thanks.

That Mac would have shipped with two OS X installation discs specific to that exact model. You will need them.

Will it function long enough for you to determine its exact model designation?

Identify your iMac model - Apple Support