ABCD is a square of side 4 inch. If each corner of square is

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Thu Sep 20, 2018 9:38 pm

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ABCD is a square of side 4 inch. If each corner of square is cut identically so that resultant structure becomes regular eight sided polygon. What is length of side of octagon?

A) [m]4/ (2-âˆš2)[/m]
B) [m]2âˆš2/ (âˆš2+1)[/m]
C) [m]4(âˆš2-1)[/m]
D) [m]4(âˆš2+1)[/m]
E) [m]4/ (âˆš2-1)[/m]

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Source: Beat The GMAT — Problem Solving | Thu Sep 20, 2018 9:38 pm

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

ABCD is a square of side 4 inch. If each corner of square is

by fskilnik@GMATH » Fri Sep 21, 2018 8:27 am

GMATinsight wrote:ABCD is a square of side 4 inch. If each corner of square is cut identically so that resultant structure becomes regular eight sided polygon. What is length of side of octagon?

A) 4/ (2-âˆš2)
B) 2âˆš2/ (âˆš2+1)
C) 4(âˆš2-1)
D) 4(âˆš2+1)
E) 4/ (âˆš2-1)

www.GMATinsight.com

Hi GMATinsight!

First of all, congratulations for the beautiful question. ItÂ´s exactly GMAT-like at its best: clearly written and easily-solved with proper understanding/tools!

(All measures are in inches.)

\[? = x\]
\[\left\{ \begin{gathered}
2L + x = 4 \hfill \\
x = L\sqrt 2 \,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,\sqrt 2 } \,\,\,\,2L = x\sqrt 2 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,x\sqrt 2 + x = 4\]
\[\,x\left( {\sqrt 2 + 1} \right) = \,\,4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = x = \frac{4}{{\sqrt 2 + 1}} = \frac{{4\left( {\sqrt 2 - 1} \right)}}{{\left( {\sqrt 2 + 1} \right)\left( {\sqrt 2 - 1} \right)}} = 4\left( {\sqrt 2 - 1} \right)\]

The correct answer is therefore [spoiler]__(C)______[/spoiler] .

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Sat Sep 22, 2018 8:08 pm

GMATinsight wrote:ABCD is a square of side 4 inch. If each corner of square is cut identically so that resultant structure becomes regular eight sided polygon. What is length of side of octagon?

A) [m]4/ (2-âˆš2)[/m]
B) [m]2âˆš2/ (âˆš2+1)[/m]
C) [m]4(âˆš2-1)[/m]
D) [m]4(âˆš2+1)[/m]
E) [m]4/ (âˆš2-1)[/m]

www.GMATinsight.com

Refer to the image, which is self-explanatory.

We see that 4 - 2x and âˆš2x are the equal sides of the regular octagon.

Thus,

4 - 2x = âˆš2x
16 - 16x + 4x^2 = 2x^2; squaring both the sides
8 - 8x + x^2 = 0

x = [8 Â± âˆš(64 - 32)] / 2 = = [8 Â± âˆš32] / 2 = [8 Â± 4âˆš2] / 2 = 4 Â± 2âˆš2

x cannot be 4 + 2âˆš2 since x < 4, side of the square, thus x = 4 - 2âˆš2

Thus, length of the octagon = 4 -2(4 - 2âˆš2) = 4 - 8 + 4âˆš2 = 4âˆš2 - 4 = 4(âˆš2 - 1)

The correct answer: C

Hope this helps!

-Jay
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