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It is given that  all the four vertices of the rectangle lie on a circle. Let  ABCD be the rectangle ,then the chords( diagonals AC, BD of the rectangle) subtend an angle of 90°at  each of the vertices A,C and B,D respectively. Therefore,the diagonals AC,BD will be the diameter of the circle in which the rectangle ABCD is inscribed.

Let length of the rectangle = x  units and

Breadth  of the rectangle = y units

Therefore Perimeter  of the rectangle = P = 2x+ 2y = 2y*{ (x/y)+ 1).                                                               It is given that the ratio of length to breadth of the rectangle = (x/y)  = (4/3).                                     Therefore  P= 2y{(4/3)+1) =  2y(7/3) =   (14y/3).                                                                                          Now diameter of the circle d = diagonal AC= √(x^2 +y^2) = y* √{(x/y)^2 +1}=                                                   y*√{(4/3)^2 +1}  = y* √{(16/9) +1} =  y* √(25/9) = y*5/3 = 5y/3.                                                             Therefore circumference of the circle C = πd = π*(5y/3). = 5πy/3                                                               Ratio of perimeter of rectangle to circumference of circle =P/C =( 14y/3)/( 5πy/3)=14/(5π)

Answer:  P/C =  14/(5π)     Option d

 

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