Bunuel wrote:

The hexagon ABCDEF is regular. That means all its sides are the same length and all its interior angles are the same size. Each side of the hexagon is 2 feet. What is the area of the rectangle BCEF?

(A) 4 square feet

(B) \(4\sqrt{3}\) square feet

(C) 8 square feet

(D) \(4 + 4\sqrt{3}\) square feet

(E) 12 square feet

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Rectangle widthConnect sides BF and CE of the rectangle

All sides of the hexagon have length 2 feet. Width of rectangle BCEF = 2 feet

Rectangle lengthCreate two right triangles at angle D

Draw a perpendicular bisector from angle D to opposite side EC

A regular hexagon has internal angle measures of 120 degrees:

(n-2)(180) = 720 degrees total, divided by 6 angles = 120 each

Angle D is bisected into 120/2 = two angles of 60°, so

angle CDX = 60

Angle DXC = 90

Angle DCX = 30

--Either: Angle DCB = 120. Rectangle vertex BCE = 90, so (120 - 90) = 30; or

--Two of the angles in Δ DCX total 150; (180- 150) = 30

There are now two identical right 30-60-90 triangles: Δ DEX and Δ DCX

30-60-90 right triangles have side lengths in ratio \(x: x\sqrt{3}: 2x\)

Side DE, opposite the 90° angle, corresponds with \(2x\) and = \(2\)

Side DX, opposite the 30° angle, corresponds with \(x\) and therefore = \(1\)

Side EC, opposite the 60° angle, corresponds with

\(x\sqrt{3}\) and therefore

\(= \sqrt{3}\)The two right 30-60-90 triangles are identical; the length of the rectangle is the sum of their sides of \(\sqrt{3}\)

Length of rectangle:

\(2\sqrt{3}\)Area of rectangle\(Area = (L * W) = (2 * 2\sqrt{3}) = (4\sqrt{3})\) square feet

Answer B

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