How can I use Mac with ethernet in a room without ethernet wiring?

Have a look for Powerline Adapters, these allow you to connect to your router using the household mains electricity socket as if it were a hard wired Ethernet connection. Inexpensive and easy to set up. Of course your Mac would need to have an Ethernet capability. I have used TP-link adapters for a number of years.

The Router and iMac do not have to be in the same room to connect wirelessly.

see "Connect to a Wi-Fi network" at > Connect to the internet with your Mac - Apple Support

If you want a wired Ethernet connection, then you will need to run or have someone run a CAT6 or better cable Ethernet cable from the Router to the iMac.

Is it a brand new Mac? They all have built-in wi-fi so you wouldn't need any adapter.

What model of Mac (iMac, MacBook, etc) and what model year is it? Both of those things are available at Apple menu > About this Mac.

Also, it would be usual to use wi-fi unless your Mac was in a room very close to an ethernet connection (e.g., same room as a router).

I found that an attempt to print to an AirPrint printer over a Powerline adapter failed here, and succeeded when using Wi-Fi. Depending upon the noise quality of the current transmission in the household, the rating of the Powerline adapter may be optimistic.

Nope, Macs have had Wi-Fi built into them for about20 years or so.

Please click Connect to the internet with your Mac - Apple Support