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

1 Answer
Cesareo R.
May 23, 2016

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

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