Question

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

Answer 1

The length of the arc is `=1.48`

Explanation:

The angle `theta=1/3pi`

The distance between the points is

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

`=sqrt(1+1)`

`=sqrt2`

This is the length of the chord

So,

`d/2=rsin(theta/2)`

`r=d/(2sin(theta/2))`

`=sqrt2/(2sin(1/6pi))`

`=1.41`

The length of the arc is

`L=rtheta=1.41*1/3pi=1.48`

Answer 2

`s = sqrt2pi/3`

Explanation:

Begin by finding the square of the length of the chord connecting the two points:

`c^2 = (3-2)^2+(4-5)^2`

`c^2 = 1^2+(-1)^2`

`c^2 = 2`

We can use a variant the Law of Cosines:

`c^2 = a^2+b^2-2(a)(b)cos(theta)`

To find the radius of the circle, by letting, `c^2 = 2`, `a = b = r`, and `theta = pi/3`:

`2 = r^2+r^2-2(r)(r)cos(pi/3)`

`2= 2r^2-2r^2(1/2)`

`2 = r^2`

`r=sqrt2`

We know that the arclength, s, between two points on a circle is the product of the radius and the radian measure of the central angle:

`s = rtheta`

Substitute the values for `r and theta`

`s = sqrt2pi/3`