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A dart board is a regular octagon divided into regions as shown below.

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Bunuel
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A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   31 Mar 2019, 22:55
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A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

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

B. 1/4

C. \(\frac{2 - \sqrt{2}}{2}\)

D. \(\frac{\sqrt{2}}{4}\)

E. \(2 - \sqrt{2}\)

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m1033512
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Re: A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   02 Apr 2019, 08:07
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IMO A


Let the side of octagon be r
exterior angle of octagon = 360/8 = 45

So interior angle = 180 -45 = 135

So all the triangles will have 45-90-45 combinations

c sides of triangles = r/\sqrt{2}

Calculate the total area of octagon and the area of central square

Probability = area of square /area of octagon = r^2/(r^2+r^2+2\sqrt{2}+r^2)
= 1/(2+2\sqrt{2})

take 2 common and multiply and divide by \sqrt{2} -1

we get option A as answer


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A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   18 Sep 2019, 08:29
Let the side of octagon be r
exterior angle of octagon = 360/8 = 45

So interior angle = 180 -45 = 135

So all the triangles will have 45-90-45 combinations

c sides of triangles = \(r/\sqrt{2}\)

Calculate the total area of octagon and the area of central square

Probability = area of square /area of octagon = \(r^2/(r^2+r^2+2\sqrt{2}+r^2)\)
= \(1/(2+2\sqrt{2})\)

take 2 common and multiply and divide by \(\sqrt{2} -1\)

we get option A as answer

P.S: I just formatted the post of m1033512. My solution is in next comment
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Re: A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   18 Sep 2019, 08:36
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Bunuel wrote:

A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

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

B. 1/4

C. \(\frac{2 - \sqrt{2}}{2}\)

D. \(\frac{\sqrt{2}}{4}\)

E. \(2 - \sqrt{2}\)

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The attachment 7274a387184a2ce0dc35e8ecaac8b5da8b052e4c.png is no longer available


Please find attached image for my solution.

Thank you!
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menonpriyanka92
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Re: A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   20 Sep 2019, 20:45
TheNightKing wrote:
Bunuel wrote:

A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

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

B. 1/4

C. \(\frac{2 - \sqrt{2}}{2}\)

D. \(\frac{\sqrt{2}}{4}\)

E. \(2 - \sqrt{2}\)

Show Spoiler
Attachment:
7274a387184a2ce0dc35e8ecaac8b5da8b052e4c.png


Please find attached image for my solution.

Thank you!



In the last step where you multiply √2−1 in the numerator as well denominator, won't it become (√2−1)^2 in the denominator and cancel out √2−1 in the numerator?
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Re: A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   20 Sep 2019, 20:52
Quote:
In the last step where you multiply √2−1 in the numerator as well denominator, won't it become (√2−1)^2 in the denominator and cancel out √2−1 in the numerator?


menonpriyanka92
Yes, It would but if the terms had been \((\sqrt{2}-1)(\sqrt{2}-1)\). But they are not.
The terms in the denominator are \((\sqrt{2}+1)(\sqrt{2}-1)\)which is equal to \(a^2-b^2\)

I hope that helps!
Thank you
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A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   17 Apr 2024, 07:45
­Can such questions come in the GMAT Focus edition? (As this involves a bit of Geometry and Geometry has been removed from the syllabus for the GMAT Focus Edition)­ Bunuel
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Re: A dart board is a regular octagon divided into regions as shown below. [#permalink] New post   17 Apr 2024, 07:54
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AlishaRukhsar wrote:
­Can such questions come in the GMAT Focus edition? (As this involves a bit of Geometry and Geometry has been removed from the syllabus for the GMAT Focus Edition)­ Bunuel

Geometry is no longer part of the syllabus of the GMAT, so no.
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Re: A dart board is a regular octagon divided into regions as shown below. [#permalink]
17 Apr 2024, 07:54
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