Q: Multiple screens on mid-2007 iMac with Mavericks os

Hello Peartree327,

I understand that you are trying to get your iMac to have a TV as it's secondary display. Let's go over what you can do to get that working.

That port for video on the right is going to be a Mini-DVI port. You would pair that plus a DVI to HDMI cable, you should be able to get video to your TV. Take a look at the articles below to go over the tech specs and information about the adapters to get you sorted out. 

iMac (Mid 2007) - Technical Specifications
https://support.apple.com/kb/sp16

About Apple video adapters and cables
https://support.apple.com/en-us/HT201853​

Apple Mini-DVI to DVI Adapter

Part: M9321G

Compatible with: 

• MacBook (13-inch, Mid 2009)
• MacBook (13-inch, Early 2009)
• MacBook (13-inch, Late 2008)
• MacBook (13-inch, Early 2008)
• MacBook (13-inch, Late 2007)
• MacBook (13-inch, Mid 2007)
• MacBook (13-inch, Late 2006)
• MacBook (13-inch)
• Mac mini (Late 2009)
• Mac mini (Early 2009)
• iMac (Early 2008)
• iMac (Mid 2007)
• iMac (Late 2006)
• iMac (Early 2006)

Belkin HDMI to DVI-D Cable (2 m/6.6 ft.)
http://www.apple.com/shop/product/HHM62ZM/A/belkin-hdmi-to-dvi-d-cable-2-m-66-ft  

Take care.