How is the Distance Formula the same as the Pythagorean Theorem?

Question

How is the Distance Formula the same as the Pythagorean Theorem? No matter how I try to see it, I can't find any similarities.

Answer 1

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Take 2 points (x1,y1) and (x2,y2) and graph them. Draw the line between them and call its length c. Now draw a line from (x1,y1) to (x2,y1) and call its length a, and draw a line from (x2,y1) to (x2,y2) and call its length b.

a is equal to x2-x1. b is equal to y2-y1. The Pythagorean theorem tells us that c^2=a^2+b^2, so:

c^2 = (x2-x1)^2 + (y2-y1)^2

c = sqrt[(x2-x1)^2 + (y2-y1)^2]

which is the distance formula.

Answer 2

The distance formula is the formula to find the the distance between two points given their Cartesian co-ordinates and the assumption that they lie on a flat plane.

The two links below explain it with diagrams. Its actually a straightforward application of the Pythagoras theorem.

If the points are expressed geographically so that we have northings (N1, N2) and eastings (E1 and E2) , the two points are E1, N1 and E2,N2.

Point 1 is (N1-N2) units north of point 2 and (E1-E2) units east of point 2.

On a flat plane north is at right angles to east so we have a right angled triangle with the distance between the points being the hypotenuse, and the north and east being the other sides.

Therefore Distance = Square root( (N1-N2)^2+(E1-E2)^2)

The term distance formula is also sometimes used for the formula:

Distance = Rate x Time

This formula has nothing to do with Pythagoras.

Source(s): http://math.about.com/library/bldistance.htm http://www.purplemath.com/modules/distform.htm