Question

What is another way to find the distance between two points on a coordinate plane besides the Pythagorean theorem?

Answer 1

The other way is to use a very accurate ruler, and measure the
distance between the two points on the paper.  Alternatively, if
you have it on the computer, you can blow up the image on the
screen and measure it very precisely.

If you don't want to measure it, and you want to use the coordinates
of the points to calculate it, then any method you use is going to boil
down to the Pythagorean Theorem.