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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