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