Question

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

Answer 1

`2.342\ \text{ unit`

Explanation:

The distance `d` between the given points `(5, 2)` & `(4, 4)` is

`d=\sqrt{(5-4)^2+(2-4)^2}`

`=\sqrt{5}`

Now, the radius `R` of circle is given as

`\sin ({5\pi}/6)=\frac{d/2}{R}`

`\sin(\pi/6)=d/{2R}`

`1/2=\sqrt5/{2R}`

`R=\sqrt5`

hence, the shortest arc length between given points which are `{5\pi}/3` apart

`=(2\pi-{5\pi}/3)R`

`=\pi/3(\sqrt5)`

`=2.342\ \text{ unit`