could someone help

Question

Give an in depth explanation about how the Pythagorean Theorem and the Distance Formula are similar.

 

 

Answer 1

In the Cartesian Coordinate System, the way to find the distance between two points,

A (x₁,y₁) and B (x₂, y₂), is to consider AB to be the hypotenuse of the right triangle:

The other point C is (x₂,y₁).

|y₂....................B

|.............c.../....| b

|............../.......|

y₁.....A/_.._a_.._C

|

|___x₁_____x₂________

The Pythagorean Theorem is:

c² = a² + b²

or

c = SQRT (a² + b²):

AB = SQRT((AC)² + (AB)^{2 )}

AB = SQRT ( (x₂-x₁)² - (y₂-y₁)² )

So, the Pythagorean Theorem is written in terms of c². The distance formula is written in terms of c. They are the same equation (one has both sides squared).

Answer 2

The general distance formula for any 2 points is d = sqrt((x₁-x₂)²+(y₁-y₂)²). In the case of a right triangle, if we want to find the length of the hypotenuse, we can revolve and/or reflect the triangle so that it is wholly within the 1st quadrant and the right angle vertex C is at (0,0). Then, the other 2 vertices A and B would be at (0,y) and (x,0) respectively . Then by the general formula, the 3 sides would have lengths of:

d(AC) = sqrt((0-0)²+(y-0)²) = sqrt(y²) = y

d(BC) = sqrt((x-0)²+(0-0)²) = sqrt(x²) = x

d(AB) = sqrt((x-0)²+(y-0)²) = sqrt(x²+y²) = c (length of hypotenuse)

So if we square both sides (which we can do because of the equality), we get c² = x² + y², which is the Pythagorean Theorem.