Question

how can you use the pythagorean theorem to solve real-world problems

Answer 1

Painters use ladders to paint on high buildings, and they use Pythagoras theorem to complete their work. 

Answer 2

The Pythagorean theorem is used in most physics problems for vectoring as well as being incredibly important in proving calculus, the Pythagorean theorem is likely one of the most critical things ever put together and most later math is based heavily upon it.