Bunuel wrote:

A regular polygon is inscribed in a circle. How many sides does the polygon have?

(1) The length of the diagonal of the polygon is equal to the length of the diameter of the circle.

(2) The ratio of area of the polygon to the area of the circle is less than 2:3.

Question : How many sides does the Regular polygon have?Statement 1: The length of the diagonal of the polygon is equal to the length of the diameter of the circle.In a polygon of n sides, (n-3) diagonals can be drawn from every vertex of the polygon and all the diagonals will not be of the same length However, Since the length of the diagonal of the polygon is equal to the length of the diameter of the circle therefore we understand that the longest diagonal is passing through the centre and hence the number of sides of the polygon will be even but the no. of sides of polygon may be 4 or 6 or 8

Hence,

NOT SUFFICIENTStatement 2: The ratio of area of the polygon to the area of the circle is less than 2:3Case 1: Square inside a Circle or radius=1Area of Circle = (Pi)r^2 = 3.14 x 1 = 3.14

Area of Square = Side^2 = \([(Diameter of Circle/\sqrt{2})]^2\)= \([2/\sqrt{2}]^2\) = \((\sqrt{2})^2\) = 2

Area of Square / Area of Circle = 2/3.14 i.e. Less than 2/3

Case 1: Equilateral Triangle inside a Circle or radius=1Area of Circle = (Pi)r^2 = 3.14 x 1 = 3.14

Area of Equilateral Triangle inside that circle will be less than the Area of Square inside the circle as calculated in case 1

Area of Equilateral Triangle / Area of Circle will be Less than 2/3

So the Number of Sides of the Regular Polygon may be either 3 or 4

Hence,

NOT SUFFICIENTCombining the Two statements:The Number of sides of the Polygon must be even i.e. n= 4 or 6 or 8 etc.

The case of Equilateral Triangle can be ruled out as there is no Diagonal and it has odd number of sides

and

The case of Square is acceptable as per both the Statements

and

The case of Hexagon will result in the ratio of area of Hexagon and Area of Circle as Greater than 2/3 hence not acceptable

Similarly The case of Polygon with more than 4 sides will result in the ratio of area of Polygon and Area of Circle as Greater than 2/3 hence not acceptable

Thus only SQUARE is the possible Polygon hence

SUFFICIENTAnswer: Option

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