The fifth badge is in Sootopolis City, you need Surf, Dive, and Whirlpool to get it
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In Pokekon Sapphire, you get the 5th badge after defeating your opponent in a Gym Battle at Petalburgh.
go to your mom after getting your 5th badge this works for both sapphire and ruby but am not sure about emerald !!!!!!!!!
the third badge in sapphire is at mauville city(north of slateport city)
After you get the 4th badge go to petalburg city and enter the gym go through it then fight Norman for your 5th badge.
in lavarage town