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# Square PQRS is inscribed in the square ABCD whose

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Square PQRS is inscribed in the square ABCD whose [#permalink]

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05 Mar 2014, 18:23
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Square PQRS is inscribed in the square ABCD whose perimeter is four. What is the area of the shaded region:

A: $$\frac{1}{12}$$

B: $$\frac{\sqrt{2}}{8}$$

C: $$\frac{1}{16}$$

D: $$\frac{1}{8}$$

E: $$\frac{\sqrt{2}}{16}$$
[Reveal] Spoiler: OA

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Last edited by PareshGmat on 30 Jul 2014, 17:43, edited 2 times in total.

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Re: Square PQRS is inscribed in the square ABCD whose [#permalink]

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05 Mar 2014, 21:36
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PareshGmat wrote:
Square PQRS is inscribed in the square ABCD whose perimeter is four. What is the area of the shaded region:

A: $$\frac{1}{12}$$

B: $$\sqrt{2}/8$$

C: $$\frac{1}{16}$$

D: $$\frac{1}{8}$$

E: $$\sqrt{2}/16$$

There are various ways of approaching it.

Method 1:
Area of ABCD is 1.
Area of PQRS = (1/2)*(diagonal1)*(diagonal2)
Both diagonals of PQRS are 1 each since they are the same lengths as sides of ABCD
Area of PQRS = (1/2)*1*1 = 1/2

Area of leftover region = 1 - 1/2 = 1/2
The leftover region after you cut out PQRS is split into 8 equal areas and the red region is one of those 8.
Hence area of red region is (1/8)*(1/2) = 1/16

Method 2:

The perimeter of ABCD is 4 so each side is 1. So each half side is 1/2.
In triangle APQ, AP and AQ are 1/2 each so $$PQ = 1/\sqrt{2}$$ (using Pythegorean theorem)
So half of PQ is $$1/2\sqrt{2}$$

Area of red triangle = $$(1/2)*(1/2\sqrt{2})*(1/2\sqrt{2}) = 1/16$$
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18148 [2], given: 236 VP Joined: 02 Jul 2012 Posts: 1214 Kudos [?]: 1721 [0], given: 116 Location: India Concentration: Strategy GMAT 1: 740 Q49 V42 GPA: 3.8 WE: Engineering (Energy and Utilities) Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 05 Mar 2014, 21:38 Side of outer square = 1 Half of side of outer square = 0.5 Area of outer square = 1 Area of inner square = $$(side of inner square)^2$$ $$(side of inner square)^2= 0.5^2 + 0.5^2 = 0.25 + 0.25 = 0.5$$ Area of outer square - area of inner square = 1 - 0.5 = 0.5 Shaded area = $$\frac{1}{8}(0.5) = \frac{1}{16}$$ Answer is C _________________ Did you find this post helpful?... Please let me know through the Kudos button. Thanks To The Almighty - My GMAT Debrief GMAT Reading Comprehension: 7 Most Common Passage Types Kudos [?]: 1721 [0], given: 116 SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1848 Kudos [?]: 2794 [0], given: 193 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 05 Mar 2014, 23:30 Area of ABCD = 1 Area if inscribed square PQRS $$= \frac{1}{2}$$ (Its always half) Look at the diagram attached (Green region 4 triangles) As the area of inscribed square PQRS $$= \frac{1}{2},$$ area of each green shaded triangle $$= \frac{1}{(2*4)} = \frac{1}{8}$$ Area shaded (in red) is half of the triangle, so Answer $$= \frac{1}{16}$$ Attachments sq.jpg [ 30.48 KiB | Viewed 21250 times ] _________________ Kindly press "+1 Kudos" to appreciate Kudos [?]: 2794 [0], given: 193 Math Expert Joined: 02 Sep 2009 Posts: 42646 Kudos [?]: 135935 [1], given: 12716 Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 06 Mar 2014, 00:53 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED PareshGmat wrote: Square PQRS is inscribed in the square ABCD whose perimeter is four. What is the area of the shaded region: A: $$\frac{1}{12}$$ B: $$\sqrt{2}/8$$ C: $$\frac{1}{16}$$ D: $$\frac{1}{8}$$ E: $$\sqrt{2}/16$$ Minimum calculations are necessary to solve this question. Consider the diagram below: Attachment: Untitled.png [ 6.72 KiB | Viewed 21209 times ] The red region is 1/4 th of 1/4 th of the big square, so 1/16 th of the big square. Answer: C. _________________ Kudos [?]: 135935 [1], given: 12716 Math Expert Joined: 02 Sep 2009 Posts: 42646 Kudos [?]: 135935 [1], given: 12716 Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 06 Mar 2014, 01:16 1 This post received KUDOS Expert's post Bunuel wrote: PareshGmat wrote: Square PQRS is inscribed in the square ABCD whose perimeter is four. What is the area of the shaded region: A: $$\frac{1}{12}$$ B: $$\sqrt{2}/8$$ C: $$\frac{1}{16}$$ D: $$\frac{1}{8}$$ E: $$\sqrt{2}/16$$ Minimum calculations are necessary to solve this question. Consider the diagram below: Attachment: Untitled.png The red region is 1/4 th of 1/4 th of the big square, so 1/16 th of the big square. Answer: C. Similar questions on "shaded regions" to practice. PS: the-shaded-region-in-the-figure-above-represents-a-135095.html in-the-figure-shown-if-the-area-of-the-shaded-region-is-3-t-104668.html regular-hexagon-abcdef-has-a-perimeter-of-36-o-is-the-cente-89544.html the-rectangular-region-above-contains-two-circles-and-a-semi-161428.html in-the-figure-given-below-abcd-is-a-square-and-p-q-r-and-160941.html abc-is-an-equilateral-triangle-of-area-3-and-arc-de-is-cent-160282.html if-abcd-is-a-square-with-area-625-and-cefd-is-a-rhombus-wit-105631.html the-area-of-each-of-the-16-square-regions-in-the-figure-abov-159463.html triangle-abo-is-situated-within-the-circle-with-center-o-so-151050.html four-identical-circles-are-drawn-in-a-square-such-that-each-156620.html in-the-diagram-points-a-b-and-c-are-on-the-diameter-of-127285.html in-the-xy-plane-a-triangle-has-vertexes-0-0-4-0-and-88395.html the-shaded-portion-of-the-rectangular-lot-shown-above-repres-144379.html the-triangles-in-the-figure-above-are-equilateral-and-the-62201.html in-the-figure-above-triangles-abc-and-mnp-are-both-isosceles-127532.html the-shaded-region-in-the-gure-above-represents-a-circular-129941.html abcd-is-a-square-picture-frame-see-figure-efgh-is-a-127823.html h-g-f-and-e-are-midpoints-of-the-sides-of-square-abcd-127367.html arcs-de-ef-fd-are-centered-at-c-b-and-a-in-equilateral-127348.html the-figure-represents-five-concentric-quarter-circles-the-127282.html in-the-rectangular-coordinate-system-above-for-which-of-the-105212.html in-the-figure-shown-if-the-area-of-the-shaded-region-is-3-t-104668.html abc-is-a-triangle-with-area-1-af-ab-3-be-bc-3-and-ed-101041.html the-figure-shown-above-consists-of-three-identical-circles-99874.html if-the-shaded-area-is-one-half-the-area-of-the-triangle-abc-97286.html the-figure-above-represents-a-square-garden-that-is-divided-100527.html DS: what-fraction-of-the-square-region-in-the-figure-above-is-sh-155481.html points-m-and-p-lie-on-square-lnqr-and-lm-lq-what-is-the-162164.html what-is-the-area-of-the-shaded-region-above-if-abcd-165884.html equilateral-triangle-bdf-is-inscribed-in-equilateral-triangl-96109.html Hope it helps. _________________ Kudos [?]: 135935 [1], given: 12716 Senior Manager Joined: 28 Apr 2014 Posts: 272 Kudos [?]: 40 [0], given: 46 Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 30 Jul 2014, 05:56 how do we determine that P,Q,R and S are mid-points of the sides of ABCD ? Kudos [?]: 40 [0], given: 46 SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1848 Kudos [?]: 2794 [0], given: 193 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 30 Jul 2014, 17:53 himanshujovi wrote: how do we determine that P,Q,R and S are mid-points of the sides of ABCD ? Refer diagram below: Draw a square Join both diagonals of the square to get the midpoint (CG) From that midpoint, you can draw a circle with diameter same as that of the side of the square The circle will get inscribed in the square, the point of contact of that would be the midpoints of the side of the square Hope this clarifies Attachments ins.png [ 2.26 KiB | Viewed 20485 times ] _________________ Kindly press "+1 Kudos" to appreciate Kudos [?]: 2794 [0], given: 193 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7799 Kudos [?]: 18148 [0], given: 236 Location: Pune, India Re: Square PQRS is inscribed in the square ABCD whose [#permalink] ### Show Tags 30 Jul 2014, 20:05 himanshujovi wrote: how do we determine that P,Q,R and S are mid-points of the sides of ABCD ? A square is a symmetrical figure. All its sides and all angles are equal. When you inscribe another symmetrical figure in it, the final figure will also be symmetrical! Is there any reason why P should be closer to A than to D? Since all points are equivalent, P will be in the middle of A and D. You can use the same argument for all points. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Square PQRS is inscribed in the square ABCD whose [#permalink]

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30 Jul 2014, 22:32
VeritasPrepKarishma wrote:
himanshujovi wrote:
how do we determine that P,Q,R and S are mid-points of the sides of ABCD ?

A square is a symmetrical figure. All its sides and all angles are equal. When you inscribe another symmetrical figure in it, the final figure will also be symmetrical! Is there any reason why P should be closer to A than to D? Since all points are equivalent, P will be in the middle of A and D. You can use the same argument for all points.

Ok.. At the back of my mind ,I could figure out that the inherent symmetry of the figures in question would have play over here but was not able to recollect any rule or axiom from my study of geometry

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Re: Square PQRS is inscribed in the square ABCD whose [#permalink]

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30 Jul 2014, 22:34
PareshGmat wrote:
himanshujovi wrote:
how do we determine that P,Q,R and S are mid-points of the sides of ABCD ?

Refer diagram below:

Draw a square
Join both diagonals of the square to get the midpoint (CG)
From that midpoint, you can draw a circle with diameter same as that of the side of the square
The circle will get inscribed in the square, the point of contact of that would be the midpoints of the side of the square

Hope this clarifies

Sorry but not able to get this logic. karishma's comment is much more intuitive and I could feel it but could not take it as a necessary pre-requisite in this question

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Re: Square PQRS is inscribed in the square ABCD whose [#permalink]

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30 Jul 2014, 23:18
himanshujovi wrote:
PareshGmat wrote:
himanshujovi wrote:
how do we determine that P,Q,R and S are mid-points of the sides of ABCD ?

Refer diagram below:

Draw a square
Join both diagonals of the square to get the midpoint (CG)
From that midpoint, you can draw a circle with diameter same as that of the side of the square
The circle will get inscribed in the square, the point of contact of that would be the midpoints of the side of the square

Hope this clarifies

Sorry but not able to get this logic. karishma's comment is much more intuitive and I could feel it but could not take it as a necessary pre-requisite in this question

Never mind... What I meant to say is if you rotate the inscribed square, it will follow the circle & the circle will TOUCH the outer square only on its midpoints

In this way PQRS are the respective midpoints
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Re: Square PQRS is inscribed in the square ABCD whose [#permalink]

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07 Jan 2016, 09:43
VeritasPrepKarishma wrote:
himanshujovi wrote:
how do we determine that P,Q,R and S are mid-points of the sides of ABCD ?

A square is a symmetrical figure. All its sides and all angles are equal. When you inscribe another symmetrical figure in it, the final figure will also be symmetrical! Is there any reason why P should be closer to A than to D? Since all points are equivalent, P will be in the middle of A and D. You can use the same argument for all points.

Unless stated in the question (or maybe I'm misinterpreting the meaning of 'inscribed'), there is actually no reason that the inscribed square should touch on the midpoints of the outer square. Consider the figure below:

The inner square only touches the outer square at 4 points, but they are clearly not the midpoints of the sides of the outer square. In this case, we would not be able to figure out the area of the shaded region without additional information (like the ratio of AP to PD). Likewise, for the problem in question, unless we are given more information to know that the inscribed square touches the outer square at its midpoints, then we cannot solve the problem without making assumptions, something we should rarely ever do on the GMAT.

What is the source of the question? It's certainly not an official GMAT problem...
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Re: Square PQRS is inscribed in the square ABCD whose [#permalink]

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