Question

Points `(5 ,2 )` and `(1 ,4 )` are `(5 pi)/3` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

Arc Length, `L = r theta = 2sqrt(5/3)*pi/3 = (2pi)/3*sqrt(5/3)`

Explanation:

Take a look at the points in the circle in the figure:
The chord `bar(AB)` can be calculated the distance formula:
`AB = sqrt((5-1)^2 + (2-4)^2) = sqrt(16+4)=sqrt(20)=2sqrt(5)`
Now we know the outside angle between `bar(OB)` and `bar(OA)`
angle `alpha=BhatOA = (5pi)/3` thus the inside angle `theta=BhatOA = (pi)/3`
From the picture we can see that `bar(BO) = bar(AO) = r = Radius`
Also `bar(AB) = bar(AH) + bar(HB) = sqrt(5)`

Now triangle `AOH` form a right angle triangle with angles:
`/_HAO = beta = pi/3`

`/_AOH = theta/2 = pi/6`

This the 30-60-90 right angle triangle with sides relationship of
`(2:1 :sqrt(3))`
Using `(AOH)_(Delta)` we write `bar(AO) = r = bar(AH)/sin(beta) = sqrt(5)/sinbeta`
Now `=>sinbeta = sqrt(3)/2`
Thus `r = 2 sqrt(5)/sqrt(3)= 2sqrt(5/3)`
Now Arc Length, `L = r theta = 2sqrt(5/3)*pi/3 = (2pi)/3*sqrt(5/3)`