How to Enter Diagnostics i-Mac 27" intel High Sierra

This is the way:

Use Apple Diagnostics to test your Mac - Apple Support

Press and hold D or Option-D.

If you're using a wireless keyboard, you should get your hands on a usb keyboard and try with that. Sometimes a wireless keyboard doesn't respond quickly enough in situations like this.

Apple Diagnostics (AD) was not available on Macs until AFTER June 2013:

How to use Apple Hardware Test (AHT) on your Mac - Apple Support

but 27-inch iMacs have been available since late 2009. That is why D.I. Johnson's "exactly which" questions are so important to our helping here where we cannot see nor touch your tech gear.

If the hard drive has been erased or replaced, embedded AD or AHT has been lost and your need RECOVERY MODE, not AD or AHT, and an internet connection. See:

Use macOS Recovery on an Intel-based Mac - Apple Support

Coaxing a Mac to boot to diagnostics (or recovery) can be tricky.

Either press the key when you press the power button to restart, or press when you hear the startup chime. I find the latter consistently works for me.

Continue to hold until you see the Apple logo and progress bar. You may also continue to hold until the computer finishes booting, but that's not usually required.

Exactly which Mac model and year are you using, and which macOS version?

Has the Hard Drive in this iMac been completely wiped or replaced...?

If so, then the on-board test may no longer be present.

Likewise certain models, will need to run the Apple Hardware Test from the internet or from install disc 2.

see "Learn More" at > How to use Apple Hardware Test on your Mac - Apple Support

The Late 2009 iMac came with a gray Snow Leopard Install Disc set and the Hardware Test is on Disc 2.