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In the figure above, the equilateral hexagon \(ABCDEF\) is inscribed in the circle with center \(G,\) which has a diamet

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by Gmat_mission » Thu Sep 10, 2020 1:02 am

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2018-11-13_1220.png
In the figure above, the equilateral hexagon \(ABCDEF\) is inscribed in the circle with center \(G,\) which has a diameter of \(16.\) What is the length of \(AB?\)

A. \(16\)

B. \(8\sqrt3\)

C. \(6\sqrt3\)

D. \(8\)

E. \(4\sqrt3\)

Answer: D

Source: Princeton Review
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Re: In the figure above, the equilateral hexagon \(ABCDEF\) is inscribed in the circle with center \(G,\) which has a di

by Nirav Umaretiya » Sat Sep 12, 2020 9:11 pm
This PS tests if you know the sum of all angles of a hexagon.
The formula for the sum of angle of an n-sided polygon is = 180 (n-2)

So the sum of the angles of the hexagon is 180 (6-2) = 180 (4)

Now, since it is an equilateral polygon, each of its angles is going to be equal.
Thus \(\frac{180\ \left(4\right)}{6}\) = 120

Now the diameter joining any two points of the hexagon will be an angle bisector (divide angle in 2 equals)

So there will be 3 diameters, creating 6 equilateral Triangles.

Equilateral triangles have equal angles and equal lengthed sides.
The 2 sides of the triangle are radius so \(\frac{16}{2}\) = 8

The third side of the triangle is the side of the hexagon, which is also going to be 8.

The correct answer is D
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