Question

How do you find the distance between `(-sqrt2,-sqrt2)`, `(sqrt2, 6sqrt2)`?

Answer 1

`abs((-sqrt2,-sqrt2):(sqrt2,6sqrt2))=color(red)(sqrt(106)`

Explanation:

The easiest way to do this is to observe that the given points are equivalent to `(-1,-1)` and `(1,6)` scaled by a factor of `sqrt(2)`

The distance between the given points will be the same as the distance between `(-1,-1)` and `(1,6)` multiplied by `sqrt(2)`

The horizontal distance between `(-1,-1)` and `(1,6)` is `1-(-1)=2`

The vertical distance between `(-1,-1)` and `(1,6)` is `6-(-1)=7`

The diagonal distance between `(-1,-1)` and `(1,6)` is given by the Pythagorean Theorem as
`color(white)("XXX")d=sqrt(2^2+7^2)=sqrt(4+49)=sqrt(53)`

and the corresponding diagonal distance between `(-sqrt(2),-sqrt(2))` and `(sqrt(2),6sqrt(2))` is
`color(white)("XXX")sqrt(2)d=sqrt(2)*sqrt(53)=sqrt(106)`