Refer :

Circle initial : Initial diagram with extra variables for explanation.

Circle Solution : Diagram with all angels and sides marked

Note: Here m writing full explanation so solution might look lengthy but when we solve without writing all the steps it can be done quickly.

Given: ABCD is a rectangle, inscribed in a circle, area 40

=> By figure : AB||CD and AC|| BD

k+n=90 and m+j=90.

Let length and breadth be l and b . so area =40 = l*b

As cord AD and BC make 90 degree at circle. So AD and CB are diagonals. At O they bisect each other. So OA=OB=OC=OD

length of cord AD and BC = \(\sqrt{l^2 +b^2}\)

OC=OD = \(\frac{\sqrt{l^2 +b^2}}{2}\)

Find: Length of Arc CD

=> Arc length = \(2*pi*r* (\frac{angle at center}{360})\)

=> Arc length = \(2*pi*OD* (\frac{angle COD}{360})\)

=> We just need to find length OD and Angle CODOption 2: The sum of the measures of angle j and angle n is 90 degrees.

so j+n=90

But As given ABCD is a rectangle. Angle D =90. And in triangle CBD sum of all angles =180

=> j+n+90=180 => j+n =90. We know this equation just by question stem. So this option doesn't give us any additional detail.

So Not sufficient.

And this information is already given in question combining 1 and 2 option will not give us any additional detail.

So our Answer is either A or E. Option 1: The sum of the measures of angle m and angle n is 60 degrees.

m+n=60.

As AB || CD => angle m=n => m=n=30

=> k=j=60

=> by figure: as OC =OD=> n=x=30 and OD=OB => j=y=60

=> angle COD = 120 and Angle DOB=60.Now as Triangle DOB becomes equilateral triangle. All sides must be equal. => OB=OD=BD

=> \(\frac{\sqrt{l^2 +b^2}}{2} = b\) => \(l^2 +b^2 =4b^2\) => \(l^2=3b^2\) => \(l=\sqrt{3}b\)

as l*b=40 => \(\sqrt{3}b * b = 40\) => \(b^2= \frac{40}{\sqrt{3}}\) => \(b = \sqrt{\frac{40}{\sqrt[]3}}\)

Also as radius of circle = r = OD = \(\frac{\sqrt{l^2 +b^2}}{2}\) = \(\frac{\sqrt{3b^2 +b^2}}{2}\)

\(r = \frac{2b}{2} = b = \sqrt{\frac{40}{\sqrt[]3}}\)

So we got value of angle COD = 120 and value of radius r = \(\sqrt{\frac{40}{\sqrt[]3}}\)

So we can calculate length of Arc CD =\(2*pi*r* (\frac{angle COD}{360})\)

Sufficient

Answer: A
Attachments

**File comment:** Circle with marked length and angle values. And brief of solution

Circle Solution.jpg [ 2.12 MiB | Viewed 862 times ]

**File comment:** Initial Circle diagram with additional variables for proper understanding

circle initial.PNG [ 12.91 KiB | Viewed 849 times ]