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

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

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New post 05 Nov 2017, 00:37
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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
2017-11-05_1232.png [ 5.8 KiB | Viewed 551 times ]
[Reveal] Spoiler: OA

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Kudos [?]: 135613 [0], given: 12705

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Kudos [?]: 402 [0], given: 645

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

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New post 05 Nov 2017, 07:32
Bunuel wrote:
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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:
The attachment 2017-11-05_1232.png is no longer available

Attachment:
hhhh.png
hhhh.png [ 12.04 KiB | Viewed 410 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

Answer B

Kudos [?]: 402 [0], given: 645

The hexagon ABCDEF is regular. That means all its sides are the same l   [#permalink] 05 Nov 2017, 07:32
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The hexagon ABCDEF is regular. That means all its sides are the same l

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