Question

Explain how the distance formula can prove Pythagoras' Theorem. In your

explanation use the Pythagoras Theorem and how the distance formula can be used
with a right triangle when only given the hypotenuse.

Answer 1

Given a triangle ABC, Pythagoras' Theorem shows that:

`c^2=a^2+b^2`

Thus,

`c = √(a^2+b^2)`

The distance formula, gives an equivalent expression based on two points at the end of the hypotenuse for a triangle.

`d^2 = (x_(2) -x_(1))^2 + (y_(2) -y_(1))^2`

`d = \sqrt{(x_(2)-x_(1))^2 + (y_(2) -y_(1))^2 }`

Therefore when given the hypotenuse with endpoints at

`(x_(1), y_(1)) and {(x_(2), y_(2))`

We know that the third point of the right triangle will be at

`(x_(2), y_(1))`

and that the two side lengths will be defined by the absolute values of:

`(x_(2) - x_(1)) = a`

`(y_(2) - y_(1)) = b`