In the above figure, let OB = OC = R. Let, OE = y, AB = 2a
Now, R = x + y ...(i)
R + y = 2a ...(ii)
In triangle OEB, R^2 = y^2 + a^2 ...(iii)
Substituting (i) in (iii), we get
(x+y)^2 = y^2 + a^2
x^2 + 2xy + y^2 = y^2 + a^2
x^2 + 2xy = a^2 ...(iv)
Substituting (i) in (ii), we get
x+2y = 2a
Multiplying above equation by x, we get
x^2 + 2xy = 2ax ... (v)
From (iv) and (v), we get
a^2 = 2ax
a=0 or a=2x
Since, a is a length, it can not be non - positive.
Therefore, a = 2x ... (vi)
(1) x = 1
Substituting in (vi), we get
a = 2. Therefore, perimeter = 8a = 16.
Clearly, statement 1 is sufficient.
(2) AC and BD are perpendicular to each other
Since, ABCD is a square AC and BD are perpendicular to each other. We do not get any information about the length of the square.
Therefore statement 2 is insufficient.
Choice A is the answer.
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