Bunuel wrote:
A circle is inscribed within a regular hexagon in such a way that the circle touches all sides of the hexagon at exactly one point per side. Another circle is drawn to connect all the vertices of the hexagon. Expressed as a fraction, what is the ratio of the area of the smaller circle to the area of the larger circle?
A. √(2/3)
B. (√2)/3
C. (√3)/2
D. (√3)/4
E. 3/4
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:The first thing to recognize is that a regular hexagon has 6 equal sides and 6 equal internal angles of 120°, since the angles inside the hexagon must add up to (n – 2)×180 = 720°, or 120° per angle. This is a very symmetrical figure. The circle will touch each side exactly in the middle of the side, by symmetry, like so:
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hex-1.jpg [ 3.58 KiB | Viewed 6296 times ]
The outer circle will touch each vertex:
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hex-2.jpg [ 5.23 KiB | Viewed 6301 times ]
Now, to compare the areas of the two circles, we should compare their radii. The obvious place to draw radii is from the points of contact with the circles:
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hex-3.jpg [ 6 KiB | Viewed 6298 times ]
The triangle that’s been created is a 30-60-90 triangle. At point B, the radius is perpendicular to the side of the hexagon (which is tangent to the circle). The 120° angle of the hexagon is equally split by the longer radius, again by symmetry, creating a 60° angle at point A.
The ratios of the sides of a 30-60-90 triangle are 1: √3 : 2, with 2 as the longest side (the hypotenuse). The longer radius is the “2” side, while the shorter radius is the “√3” side.
Since the areas are proportional to the square of the radius (by A = πr^2), we know that the smaller area to the larger area would be \(\sqrt{(3)^2} : 2^2\), or 3 : 4. Expressed as a fraction, this ratio is 3/4.
Note that C is a trap answer: it expresses the ratios of the radii themselves.
The correct answer is E.
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