Question

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

Answer 1

the shortest arc happens to be `s=4.957`

Explanation:

Given:
`(5,4)` is a point on the circle
`(2,2)`is a point on the circle

Angle subtended at the center is `(5pi)/4` which is reflex lying on the major arc
The minor arc happens to be `2pi-(5pi)/4=(3pi)/4`

The triangle formed by the two points and the center happens to be an isoceles triangle with the shortest angle formed at the center being `alpha=(3pi)/4`.

Remaining angle is
`(pi-(3pi)/4)=pi/4`

Angle at each of the vertex at the base is `theta=1/2(pi)/4=pi/8`
Mid point happens to be
`((5+2)/2,(4+2)/2)=(3.5,3)`

We calculate the radius of circle by pythagoras theorem in the right angled triangle formed by one of the point, say `(5,4)`, the mid-point `(3.5,3)`, and the center of the circle.

Distance between one of the points, say `(5,4)` and the mid point `(3.5,3)` is given by
`a^2=(3.5-5)^2+(3-4)^2=(-1.5)^2+(-1)^2`
`= 2.25+1=3.25`
`a^2=3.25`

Also, the line joining `(5,4) and (3.5,3)` happens to be the adjacent side and the radius being hypotenuse for the angle at the point `(5,4)`

Consideing the ratio `costheta`,
`cos^2theta=a^2/r^2`
`theta=pi/8`
`a^2=3.25`

`cos^2(pi/8)=3.25/r^2`
`cos^2(pi)/8=0.146`
`0.854=3.25/r^2`
`r^2=3.25/0.854=3.808`
`r=sqrt3.808`
`r=1.951`

Hence, the shortest arc happens to be
`s=ralpha=1.951(3pi)/4`
`s=1.533`
the shortest arc happens to be `s=4.957`

Answer 2

Shortest Arc length `s = 4.5946`

Explanation:

`vec(AD) = sqrt(5-2)^2 + (4-2)^2) = 3.6056`

`r = (AD) / (2 sin (theta/2)) = 3.6056 / (2 sin ((5pi)/8)) = 1.95`

Since the center angle `theta` is more than `pi`, to get the shortest arc length we must subtract from `(2pi)`

Shortest length of the arc

`s = r * theta = 1.95 * (2pi - ((5pi)/4)) = 1.95 * ((3pi)/4) = 4.5946`

Answer 3

Shorter arc length between the points is `4.59` unit.

Explanation:

Angle subtended at the center by the arc is `theta_1=(5pi)/4`

Angle subtended at the center by the minor arc is

`theta=2pi-(5pi)/4= (3pi)/4 :. theta/2 =(3pi)/8`

Distance between two points `(5,4) and (2,2)` is

`D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2) =sqrt ((5-2)^2+(4-2)^2` or

`D=sqrt13 ~~ 3.61` unit. Chord length `l ~~3.61` unit.

Formula for the length of a chord is `L_c= 2r sin (theta/2)`

where `r` is the radius of the circle and `theta` is the angle

subtended at the center by the chord.

`:. 3.61=2*r*sin((3pi)/8) or r = 3.61/(2*sin((3pi)/8)` or

`r= 3.61/1.85 ~~1.95:.` Radius of the circle is `1.95` unit.

Arc length is `L_a= cancel(2 pi )* r*theta/cancel(2pi)=r *theta` or

`L_a=1.95*(3pi)/4 ~~ 4.59` unit .

Shorter arc length between the points is `4.59` unit.[Ans]