Bunuel wrote:
A circle is inscribed in a regular hexagon. A regular hexagon is inscribed in this circle. Another circle is inscribed in the inner regular hexagon and so on. What is the area of the tenth such circle?
(1) The length of the side of the outermost regular hexagon is 6 cm.
(2) The length of a diagonal of the outermost regular hexagon is 12 cm.
VERITAS PREP OFFICIAL SOLUTION:Thankfully, in DS questions, we don’t need to calculate the answer. We just need to establish the sufficiency of the given data. Note that we have found that there is a defined relation between the sides of a regular hexagon and the radius of an inscribed circle and there is also a defined relation between the radius of a circle and the side of an inscribed regular hexagon.
When the circle is inscribed in a regular hexagon,
Radius of the inscribed circle = \(\frac{\sqrt{3}}{2}\)* Side of the hexagon
When a regular hexagon is inscribed in a circle,
Side of the inscribed regular hexagon = Radius of the circle
So all we need is the side of any one regular hexagon or the radius of any one circle and we will know the length of the sides of all hexagons and the radii of all circles.
Statement I: The length of the side of the outermost regular hexagon is 6 cm.
If length of the side of the outermost regular hexagon is 6 cm, the radius of the inscribed circle is \((\sqrt{3}/2)*6 = 3\sqrt{3}\) cm
In that case, the side of the regular hexagon inscribed in this circle is also \(3\sqrt{3}\) cm. Now we can get the radius of the circle inscribed in this second hexagon and go on the same lines till we reach the tenth circle. This statement alone is sufficient.
Statement II: The length of a diagonal of the outermost regular hexagon is 12 cm.
Note that a hexagon has diagonals of two different lengths. The diagonals that connect vertices with one vertex between them are smaller than the diagonals that connect vertices with two vertices between them. Length of AC will be shorter than length of AD. Given the length of a diagonal, we do not know which diagonal it is. Is AC = 12 or is AD = 12? The length of the side will be different in the two cases. So this statement alone is not sufficient.
Answer (A).