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

1 Answer
yumean1119
Oct 28, 2017

#(pisqrt(2+sqrt(2)))/4≒1.4512#

Explanation:

Let #A(2,5), B(3,4)# and #C# the center of the circle.

#AB=sqrt((3-2)^2+(4-5)^2)=sqrt(2)#

The radius of the circle is #r=CA=CB#

Using the law of cosines leads to the equation:
#AB^2=CA^2+CB^2-2CA*CB*cos∠ACB#
#(sqrt(2))^2=r^2+r^2-2r*r*1/sqrt(2)#
#(2-sqrt(2))r^2=2#
#r^2=2/(2-sqrt(2))=2+sqrt(2)#
#r=sqrt(2+sqrt(2))#

The shorter arc length #l# between #A# and #B# is
#l=rtheta=sqrt(2+sqrt(2))*1/4pi#, or about 1.4512.

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