The distance formula makes sense in a coordinate context. In other words, they are the same thing in two seemingly different contexts. In short, the distance formula is a formalization of the Pythagorean Theorem using #x# and #y# coordinates. (depending on if #x_1 > x_2# or #x_1 < x_2#, and similarly for #y#.) Or, we could put it another way through substitutions based on the distance definitions above. 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#.

The greater the #x# contribution, the flatter the slope.

The greater the #y# contribution, the steeper the slope. Any diagonal line segment has an #x# component and a #y# component, due to the fact that a slope is #Deltay'/'Deltax#.The distance from one point to another is the definition of a line segment.There is the relationship where #sqrt((x-c)^2) = color(green)(|x-c|) = x-c ' AND ' -x+c#.The definition of a distance from #x# to #pmc# is #color(green)(|x-c|)#.We commonly write the Pythagorean Theorem as:Ĭonsider the following major points (in Euclidean geometry on a Cartesian coordinate axis): The distance formula is commonly seen as: If we consider what the distance formula really tells you, we can see the similarities.

Distance formula geometry