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05 Nov 2017, 01:37
Kudos
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
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Question Stats:
71% (02:16) correct
29%
(02:49)
wrong
based on 80
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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
Attachment:
2017-11-05_1232.png [ 5.8 KiB | Viewed 26112 times ]
05 Nov 2017, 08:32
Kudos
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Bunuel
Attachment:
hhhh.png [ 12.04 KiB | Viewed 21946 times ]
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
08 Nov 2023, 06:25
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