We have coordinates of two points:
Point A(2,8)A(2,8)
Point B(1,5)B(1,5)
Consider point OO with unknown coordinates to be a center of a circle that goes through points AA and BB.
Then we know that angle /_AOB = pi/4∠AOB=π4.
To define the length of an arc from AA to BB we need to know the radius of this circle R=OA=OBR=OA=OB. Then, knowing the central angle /_AOB=pi/4∠AOB=π4, we can easily determine the length of an arc from AA to BB as (piR)/4πR4.
To determine the radius R=OA=OBR=OA=OB, draw a perpendicular bisector to a segment ABAB. It should pass through a center of a circle OO.
Let the midpoint of a segment ABAB be point MM.
Consider triangle Delta AOM.
It's a right triangle with hypotenuse AO (that is, the radius R of a circle), cathetus AM being equal to half of a distance from A to B and an opposite acute angle /_AOM=pi/8.
AM=1/2 AB = 1/2 sqrt((2-1)^2+(8-5)^2)=sqrt(10)/2
Now we can determine
R=AO=(AM)/sin(/_AOM)=sqrt(10)/(2sin(pi/8))
To determine exact value of sin(pi/8), notice that
sin^2(pi/4)=[2sin(pi/8)Cos(pi/8)]^2=4sin^2(pi/8)[1-sin^2(pi/8)]
Let sin^2(pi/8)=x.
Then, since sin^2(pi/4)=(sqrt(2)/2)^2=1/2,
4x(1-x)=1/2
or
8x^2-8x+1=0
Choosing a smaller root of this quadratic equation,
x=(2-sqrt(2))/4
Therefore,
sin(pi/8) = 1/2sqrt(2-sqrt(2))
Hence,
R=sqrt(10)/sqrt(2-sqrt(2))=sqrt(10/(2-sqrt(2))=
=sqrt(10(2+sqrt(2))/[(2-sqrt(2))(2+sqrt(2))])=
=sqrt(10(2+sqrt(2))/2)=sqrt(5(2+sqrt(2))
The arc length is
L=piR/4=pi sqrt(5(2+sqrt(2)))/4
