Because the angle #(5pi)/4# is greater than #pi#, we know that this is NOT the smallest angle between the two points; the smallest angle is:
#angle theta=2pi-(5pi)/4#
#angle theta = (3pi)/4#
This will be the angle between the two radii that connect the two points #(4,4)# and #(7,3)#. The two radii and the chord, c, between the two points form a triangle, therefore, we can write an equation, using the Law of Cosines:
#c^2 = r^2 + r^2-2(r)(r)cos(theta)#
We know that #c^2# is the square of the distance between the two points and we know the value of #theta#:
#(7-4)^2+(3-4)^2 = r^2 + r^2-2(r)(r)cos((3pi)/4)#
Remove a common factor of #r^2#:
#3^2+(-1)^2= r^2(2-2cos((3pi)/4))#
Evaluate the cosine function:
#10= r^2(2-2(-sqrt2/2))#
Simplify:
#10= r^2(2+sqrt2)#
Multiply both sides by the conjugate:
#10(2-sqrt2)= r^2(2+sqrt2)(2-sqrt2)#
#10(2-sqrt2)= 6r^2#
Flip the equation and divide both sides by 6:
#r^2 = 5/3(2-sqrt2)#
Use the square root operation on both sides:
#r = sqrt(5/3(2-sqrt2))#
We know that the arclength, S, is the radius multiplied by the radian measure of angle:
#S = thetar#
#S = (3pi)/4sqrt(5/3(2-sqrt2))#
