How are the Pythagorean Theorem and the Distance Formula related?

If we consider what the distance formula really tells you, we can see the similarities. It is more than just a similar form.
The distance formula is commonly seen as:
##D = sqrt((x_1 – x_2)^2 + (y_1 – y_2)^2)##
We commonly write the as:
##c = sqrt(a^2 + b^2)##
Consider the following major points (in Euclidean geometry on a Cartesian coordinate axis):
The definition of a distance from ##x## to ##pmc## is ##color(green)(|x-c|)##.
There is the relationship where ##sqrt((x-c)^2) = color(green)(|x-c|) = x-c ” AND ” -x+c##
The distance from one point to another is the definition of a line segment.
Any diagonal line segment has an ##x## component and a ##y## component, due to the fact that a slope is ##Deltay”/”Deltax##. The greater the ##y## contribution, the steeper the slope. The greater the ##x## contribution, the flatter the slope.
What do you see in these formulas? Have you ever tried drawing a triangle on a Cartesian coordinate system? If so, you should see that these are two formulas relating the diagonal distance on a right triangle that is composed of two component distances ##x## and ##y##.
Or, we could put it another way through substitutions based on the distance definitions above. Let:
##x_1 – x_2 = pma##
##y_1 – y_2 = pmb##
(depending on if ##x_1 > x_2## or ##x_1 < x_2##, and similarly for ##y##.) Now what do you see? An equivalence. ##D = color(blue)(sqrt((pma)^2 + (pmb)^2)) = c = color(blue)(sqrt(a^2 + b^2))## In short, the distance formula is a formalization of the Pythagorean Theorem using ##x## and ##y## coordinates. In other words, they are the same thing in two seemingly different contexts.