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# The hexagon ABCDEF is regular. That means all its sides are the same l

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Math Expert
Joined: 02 Sep 2009
Posts: 44351
The hexagon ABCDEF is regular. That means all its sides are the same l [#permalink]

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05 Nov 2017, 01:37
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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

[Reveal] Spoiler:
Attachment:

2017-11-05_1232.png [ 5.8 KiB | Viewed 1462 times ]
[Reveal] Spoiler: OA

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The hexagon ABCDEF is regular. That means all its sides are the same l [#permalink]

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05 Nov 2017, 08:32
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

[Reveal] Spoiler:
Attachment:
The attachment 2017-11-05_1232.png is no longer available

Attachment:

hhhh.png [ 12.04 KiB | Viewed 1141 times ]

Rectangle width
Connect sides BF and CE of the rectangle

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

Rectangle length

Create 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

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The hexagon ABCDEF is regular. That means all its sides are the same l   [#permalink] 05 Nov 2017, 08:32
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