Question

Points `(2 ,5 )` and `(3 ,4 )` are `( pi)/4` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

`(pisqrt(2+sqrt(2)))/4≒1.4512`

Explanation:

Let `A(2,5), B(3,4)` and `C` the center of the circle.

`AB=sqrt((3-2)^2+(4-5)^2)=sqrt(2)`

The radius of the circle is `r=CA=CB`

Using the law of cosines leads to the equation:
`AB^2=CA^2+CB^2-2CA*CB*cos∠ACB`
`(sqrt(2))^2=r^2+r^2-2r*r*1/sqrt(2)`
`(2-sqrt(2))r^2=2`
`r^2=2/(2-sqrt(2))=2+sqrt(2)`
`r=sqrt(2+sqrt(2))`

The shorter arc length `l` between `A` and `B` is
`l=rtheta=sqrt(2+sqrt(2))*1/4pi`, or about 1.4512.