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A regular octagon ABCDEFGH has an area of one square unit. What is the

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A regular octagon ABCDEFGH has an area of one square unit. What is the  [#permalink]

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New post 20 Mar 2019, 23:15
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Question Stats:

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A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?


(A) \(1 - \frac{\sqrt{2}}{2}\)

(B) \(\frac{\sqrt{2}}{4}\)

(C) \(\sqrt{2} - 1\)

(D) 1/2

(E) \(\frac{1 + \sqrt{2}}{4}\)

Attachment:
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Re: A regular octagon ABCDEFGH has an area of one square unit. What is the  [#permalink]

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New post 21 Mar 2019, 00:42
Let each side of octagon be x

Area of octagon=2 (1+rt2)* x^2
As area given is 1 sq unit
We will get , x^2=1/(2(1+rt2)) {i}

Also to find area of rectangle ABFE, we need to find BF
For this extend AB and DC to meet at M,
Triangle CMB is a isosceles rt angle triangle, where CM=MB and BC=x
So CM =x/rt2
Similarly by extending CD and FE to meet at N we can find DN=x /rt2

Now BE=CM+CD+DN=x*(1+rt2)
And area ABEF= BE×AB=x^2(1+rt2)
Pitting value of x^2 from {i}

We get area ABEF=1/2

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A regular octagon ABCDEFGH has an area of one square unit. What is the  [#permalink]

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New post 21 Mar 2019, 00:58
Join AF & BE we see a rectangle been formed whose area would be equal to the 8 ∆ which are formed by joining opposite sides of the octagon
so area of the rectangle = sum of 8 ∆
each side of length and breadth has 4 ∆ so ratio ; 4/8 = 1/2
IMO D

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Bunuel wrote:
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A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?


(A) \(1 - \frac{\sqrt{2}}{2}\)

(B) \(\frac{\sqrt{2}}{4}\)

(C) \(\sqrt{2} - 1\)

(D) 1/2

(E) \(\frac{1 + \sqrt{2}}{4}\)

Attachment:
The attachment 50ef8af0ec716cfeea89017a030c7d57d3b40e4c.png is no longer available

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A regular octagon ABCDEFGH has an area of one square unit. What is the   [#permalink] 21 Mar 2019, 00:58
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