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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