How does the Pythagorean theorem relate to the distance formula?

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Answer 1 · 2018-02-22T07:24:33

The distance formula is derivable from Pythagoras' theorem...

Pythagoras' theorem tells us that a triangle with legs of length #a, b# and hypotenuse of length #c# satisfies:

#a^2+b^2 = c^2#

and hence:

#c = sqrt(a^2+b^2)#

Given two points #(x_1, y_1)# and #(x_2, y_2)#, consider a triangle with vertices at #(x_1, y_1)#, #(x_2, y_1)# and #(x_2, y_2)#

The side joining #(x_1, y_1)# to #(x_2, y_1)# is horizontal of length #abs(x_2 - x_1)#.

The side joining #(x_2, y_1)# to #(x_2, y_2)# is vertical of length #abs(y_2 - y_1)#

So these form two legs of a right angled triangle with hypotenuse of length:

#sqrt(abs(x_2-x_1)^2 + abs(y_2-y_1)^2) = sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

Note that the length of the hypotenuse is the distance between #(x_1, y_1)# and #(x_2, y_2)#.

So we have found:

#d"("(x_1, y_1), (x_2, y_2)")" = sqrt((x_2-x_1)^2+(y_2-y_1)^2)#