In the figure above, what is the largest rectangle that can

Question

In the figure above, what is the largest rectangle that can be inscribed in the circle?

(1) The arc AB is equal to 4.
(2) The rectangle is a square.

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Archit3110 · Answer

diagonal will be same
#1
60/360*2*pi *r = 4
r= 8*√2/pi
sufficient
diagonal of fig= 2* radius
#2
parallelogram is square so we get that sides are equal but no info about the arc or length
IMO A

Rakesh1987 · Answer

Archit3110

How did you get 60? How do you know that arc AB is making an angle of 60 degrees at the center of the circle? It is not given in the question.    Moreover, from Point 2, we know that the rectangle is a square, therefore the arc AB is making an angle of 90 degrees at center and not 60. Correct me if I am wrong.

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Archit3110 · Answer

Rakesh1987 
given in question is that what is the largest rectangle which can be inscribed , so what I did is I joined the diagonal of the figure at the center of the circle the digonal would intersect and make 4 equal parts ; 360/4 ; 60 each ; then using sector formula determined the part the which is falling under it .... supposedly even if the angle is not 60* in this case as the fig is not drawn to scale then also the statement would be sufficient to determine the radius and diameter of circle using which we can further determine the length diagonal of the figure inscribed..

Rakesh1987 · Answer

Dear  Archit3110 ,

Thanks for replying.

I have the following observation for you:
Given an arc AB, the question is asking, what is max area of a rectangle that can be inscribed in the circle (rephrasing the question as it lacs clarity).
And   Just because the circle happens to have a rectangle in it, one cant assume that it is the rectangle whose area the question wants us to find   (i.e. the rectangle with max area).   We have to assume that the figure is not drawn to scale, unless the questions says that the figure is drawn to scale, which I have never seen a question say.

The rectangle in the question is given for taking cues about the size of the circle using point 1/2/1&2 together.   In point 1) the length of arc is given, and in point 2) the characteristic of rectangle is given which gives us the angle that the arc makes at center which is 90 degrees   (360/4=90 degrees)  . Combining the two, we can calculate the radius of circle as below:
2 \pi r=4* \frac{360}{90} =16 or r= \frac{8}{\pi} . The max area of rectangle is area of square inscribed with diagonal  \frac{16}{\pi} . 
Side=  \frac{16}{\sqrt{2}\pi} . 
Area=  side^2 = \frac{128\sqrt{2}}{\pi^2}

Suppose the point 2 had said instead that "Length:breadth of the inscribed rectangle is  \sqrt{3} :1"; how would you have solved this question?? The question would be correct in this case as well.     Try it.

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Rakesh1987 · Answer

Hi nm97,

The OA is mentioned as A. Please share the solution.

In the figure above, what is the largest rectangle that can

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In the figure above, what is the largest rectangle that can

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