Step 1: Analyse Question StemArea of a circle = π * \(r^2\), where r is the radius of the circle.
To find the area of circle O, we need information about its radius or diameter.
Step 2: Analyse Statements Independently (And eliminate options) – AD / BCEStatement 1: The ratio of the area of the square circumscribed about circle O to the square inscribed in it is 2 to 1.
When a square is circumscribed about a circle – in other words, when a circle is inscribed in a square – the diameter of the circle is equal to the length of the side of the square.
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If the radius of circle O is r, the length of the side of the circumscribing square = 2r.
Therefore, area of square = \((2r)^2\) = 4\(r^2\)
When a square is inscribed in a circle, the diameter of the circle is equal to the length of the diagonal of the square.
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Since the radius of circle O is r, diameter = 2r; therefore, length of diagonal of square inscribed in circle O = 2r.
Area of a square = ½ * \((length of diagonal)^2\).
Therefore, area of inscribed square = ½ * \((2r)^2\) = ½ * 4\(r^2\) = 2\(r^2\).
The ratio of the areas of the two squares = 2 : 1. This is a fact that can be proven and Statement 1 states the same fact.
Knowing this fact will not help us find out the radius of the circle.
The data in statement 1 is insufficient to find a unique value of the area of circle O.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.
Statement 2: The ratio of the length of a side of the square circumscribed about circle O to the diameter of O is 1 to 1.
When a square is circumscribed about a circle – in other words, when a circle is inscribed in a square – the diameter of the circle is equal to the length of the side of the square.
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31st May 2022 - Post 3 - 1.png [ 6.23 KiB | Viewed 4158 times ]
If the radius of circle O is r, the length of the side of the circumscribing square = 2r.
Again, statement 2 is stating a mathematical fact. This is not sufficient to find the radius of circle O.
The data in statement 2 is insufficient to find a unique value of the area of circle O.
Statement 2 alone is insufficient. Answer option B can be eliminated.
Step 3: Analyse Statements by combiningFrom statement 1: The ratio of the area of the square circumscribed about circle O to the square inscribed in it is 2 to 1.
From statement 2: The ratio of the length of a side of the square circumscribed about circle O to the diameter of O is 1 to 1.
Both statements state already known facts, but do not provide us with the value for the radius. Circle O could be a very small circle or an extremely large one.
Even after combining the information from both statements, we do not have a unique figure and hence will not have a unique value for the area.
The combination of statements is insufficient to find a unique value of the area of circle O.
Statements 1 and 2 together are insufficient. Answer option C can be eliminated.
The correct answer option is E.