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A circle O with a radius of 4 is inscribed by a regular hexagon. What is the length of the pe

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A circle O with a radius of 4 is inscribed by a regular hexagon. What is the length of the pe

by Max@Math Revolution » Mon Jun 08, 2020 1:36 am

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[GMAT math practice question]

A circle O with a radius of 4 is inscribed by a regular hexagon. What is the length of the perimeter of the regular hexagon?
6.8PS.png
A. 16
B. 14√2
C. 16√3
D. 18√5
E. 20√6
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Re: A circle O with a radius of 4 is inscribed by a regular hexagon. What is the length of the pe

by i256 » Tue Jun 09, 2020 9:42 am
Perimeter of hexagon is = 6x

Now, as per the right angled triangle OHE, ^{x^2}=\ \left(\frac{^{ }x}{2}\right)^{2^{ }}\ +\ 4^2

Solve for x, x = 8/\sqrt{3}

Now 6x = 8\ x\ 2\sqrt{3} = 16\sqrt{3}
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Re: A circle O with a radius of 4 is inscribed by a regular hexagon. What is the length of the pe

by Max@Math Revolution » Wed Jun 10, 2020 2:33 am
=>

Since ∠EOF is 60°, we have ∠EOH = 30°.
Then triangle EOH is a right triangle with ∠O = 30°, ∠E = 60°, and ∠H = 90°.
When we put OE = x, we have HE = x/2 = 4/√3 or x = 8/√3 = 8√3/3.
Thus, the perimeter of the regular hexagon is 6·(8√3/3) = 16√3.

Therefore, C is the answer.
Answer: C
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