Question

Points `(5 ,4 )` and `(2 ,0 )` are `(5 pi)/4` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

Shortest arc length = 6.3756#

Explanation:

Chord `c = sqrt((5-2)^2 + (4-0)^2) = 5`

`c2 = 2.5`

`/_ C = (5pi)/4`
`:./_ (C/2) = (5pi)/8`

Also `(c/2) / r = sin /_ (C/2)`
`r = (c/2) / (sin /_ (C/2))`
`r = 2.5 / (sin ((5pi)/8) = 2.5 / 0.9239 = 2.7059`
Circumference of the circle `= 2*pi*2.7059 = 17.0018`

When center angle is `2pi`, arc length = circumference = 17.0018#

When the center angle is `((5pi)/4)`, shortest arc length

`a = (2pi - ((5pi)/4)*r = (2pi - ((5pi)/4)) * 2.7059 = 6.3756`