Question

Points `(2 ,4 )` and `(1 ,9 )` are `(3 pi)/4` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

`s = 6.50208`

Explanation:

The two points `p_1=(2,4)` and `p_2=(1,9)` define a segment such that the circle passing by `p_1,p_2` has the center located over the geometric line orthogonal to the segment and passing by `1/2(p_1+p_2)`.

Considering the angle with vertexes `p_1,c_o,p_2`, with `c_o` representing the circle center, we know

`hat( p_1,c_o,p_2) = theta = (3pi)/4`

and also

`norm (p_1-c_o) = norm (p_2-c_o) = r`

and also

`2 r sin(theta/2) = norm(p_1-p_2)`,

Calling `s` the arc length contained in `theta` we have also `s/r = theta`
Putting all together and solving for `s,r`

`((2 r sin(theta/2) = norm(p_1-p_2)),(s/r = theta))`

we get `s = 6.50208, r=2.75957`