Points `(7 ,1 )` and `(8 ,9 )` are `(3 pi)/4` radians apart on a circle. What is the shortest arc length between the points?

Distance between `(7,1)` and `(8,9)` is given by the Pythagorean Theorem as
`color(white)("XXX")sqrt(1^2+8^2)=sqrt(65)`

To determine the arc length we will need to determine the radius of a circle with the given points at an angle of `(3pi)/4` relative to the center of the circle.

(Note this diagram is not accurate).

If we let the radius of this circle be `color(green)(r)`
and denote
`color(white)("XXX")(7,1)` as `P`,
`color(white)("XXX")(8,9)` as `Q`,
`color(white)("XXX")`the center of the circle as `C`, and
`color(white)("XXX")`the point on the extension of `PC` to form a right angle with `Q` as `S`

then
Since
`color(white)("XXX")/_PSQ=pi/2`, and
`color(white)("XXX")/_QCS=pi-(3pi)/4=pi/4`
`rarr`
`color(white)("XXX")/_CQS=pi/4` and
`color(white)("XXX")abs(CS)=abs(QS)=r/sqrt(2)`

Therefore
`color(white)("XXX")abs(PS)=r+r/sqrt(2)=((sqrt(2)+1)/sqrt(2))r`
and
`color(white)("XXX")abs(PQ)=sqrt((r/sqrt(2))^2+(((sqrt(2)+1)/(sqrt(2)))r)^2)`
`color(white)("XXX")=(sqrt(2+sqrt(2)))r`

But we previously determined that
`color(white)("XXX")abs(PQ)=sqrt(65)`
So
`color(white)("XXX")r=sqrt(65)/(sqrt(2+sqrt(2))`

The shortest arc length of an arc with radius `color(green)(r)` and an angle of `(3pi)/4` is `3/4r`

`rArr` shortest arc length is `3/4xxsqrt(65)/(sqrt(2+sqrt(2)))`

`color(white)("XXX")=(3sqrt(65))/(4sqrt(2+sqrt(2)))`