Will the Radeon HD 7870/7850 work with my M5A97 AMD-based Motherboard?
Definitely. Pretty much any PCI-Express-based GPU will work on that motherboard. Heck, you could even put more than one of the GPUs on the motherboard for some CrossFireX action. The board supports quad-GPU action, so you could actually put four - yes FOUR - GPUs on that motherboard.
So you could choose any Radeon HD 7850/7870 GPU on the market, the most powerful, the most overclocked, or the best-looking, and it'll fit and work beautifully on your ASUS M5A97 motherboard.