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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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17 Jul 2008, 11:36
Here is a solution that works pretty quickly, but I'm not sure about finishing it to come up with an actual ratio of 2:1. See Picture Attached (same as other one, but this post is on page 2, so i attached it again). We're looking for BEC:ADB. Right now we have BEC = \(\frac{y^2}{2}\) and ADB = \(1*(1y)*0.5\) or \(\frac{1y}{2}\) This is the same as \(\frac{y^2}{2}\) divided by \(\frac{1y}{2}\) or \(\frac{y^2}{2}\) * \(\frac{2}{1y}=\frac{y^2}{1y}\) We know that y is a fraction because we made the entire side = 1, but how much of that side is y? Attachment:
TriangleInscribed.jpg [ 16.13 KiB  Viewed 8130 times ]



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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Updated on: 17 Jul 2008, 13:47
jallenmorris wrote: Here is a solution that works pretty quickly, but I'm not sure about finishing it to come up with an actual ratio of 2:1. See Picture Attached (same as other one, but this post is on page 2, so i attached it again). We're looking for BEC:ADB. Right now we have BEC = \(\frac{y^2}{2}\) and ADB = \(1*(1y)*0.5\) or \(\frac{1y}{2}\) This is the same as \(\frac{y^2}{2}\) divided by \(\frac{1y}{2}\) or \(\frac{y^2}{2}\) * \(\frac{2}{1y}=\frac{y^2}{1y}\) We know that y is a fraction because we made the entire side = 1, but how much of that side is y? Attachment: TriangleInscribed.jpg I guess we have to show a relationship between AD, DB, and AB such as \((AD)^2 + (DB)^2 = (AB)^2\) just as how we've established that relationship between BE, EC, and BC such as \((BE)^2 + (EC)^2 = (BC)^2\) but that will consume time.
Originally posted by tarek99 on 17 Jul 2008, 11:46.
Last edited by tarek99 on 17 Jul 2008, 13:47, edited 1 time in total.



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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17 Jul 2008, 12:18
I think I have it figured out: If area of Tri. BEC = \(\frac{y^2}{2}\) and Tri.ADB = \(\frac{1(1y)}{2}\) then we have \(\frac{y^2}{2} * \frac{2}{1y} = \frac{y^2}{1y}\) but we have to figure out what this is in numbers. So Notice that the hypotnuse of ADB and BEC are both Z so... \(y^2 + y^2 = z^2\) and \(1^2 + (1y)^2 = z^2\)..so \(y^2 + y^2 = 1^2 + (1y)^2\) \(2y^2 = 1 + 1  2y + y^2\) \(2y^2 = 2  2y + y^2\) \(y^2 = 2  2y\) \(y^2 = 2(1  y)\)  now multiply both sides by 1/2 \(\frac{y^2}{2} = 1  y\)  now multiply both sides by \(\frac{1}{y^2}\) \(\frac{1}{2} = \frac{1  y}{y^2}\) Notice above, that the ratio is \(\frac{y^2}{1y}\), so if \(\frac{1y}{y^2}=\frac{1}{2}\), then the inverse \(\frac{y^2}{1y}= 2\) or answer C!! Wow, and it only took us all day! Attachment:
TriangleInscribed.jpg [ 16.13 KiB  Viewed 2697 times ]



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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17 Jul 2008, 14:06
jallenmorris wrote: I think I have it figured out: If area of Tri. BEC = \(\frac{y^2}{2}\) and Tri.ADB = \(\frac{1(1y)}{2}\) then we have \(\frac{y^2}{2} * \frac{2}{1y} = \frac{y^2}{1y}\) but we have to figure out what this is in numbers. So Notice that the hypotnuse of ADB and BEC are both Z so... \(y^2 + y^2 = z^2\) and \(1^2 + (1y)^2 = z^2\)..so \(y^2 + y^2 = 1^2 + (1y)^2\) \(2y^2 = 1 + 1  2y + y^2\) \(2y^2 = 2  2y + y^2\) \(y^2 = 2  2y\) \(y^2 = 2(1  y)\)  now multiply both sides by 1/2 \(\frac{y^2}{2} = 1  y\)  now multiply both sides by \(\frac{1}{y^2}\) \(\frac{1}{2} = \frac{1  y}{y^2}\) Notice above, that the ratio is \(\frac{y^2}{1y}\), so if \(\frac{1y}{y^2}=\frac{1}{2}\), then the inverse \(\frac{y^2}{1y}= 2\) or answer C!! Wow, and it only took us all day! Attachment: TriangleInscribed.jpg yeah, I just got the answer the same way you did. wow, this is such a ridiculous problem. It also took me the whole day just dedicating to this one annoying question. can you imagine seeing a question like that on the real gmat??? This question was provided by the GMATprep, so you better take this question seriously! heheh....great job! now I can go to bed so that I can dream about this horrifying experience....heheh...



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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17 Jul 2008, 15:02
jallenmorris wrote: Attachment: TriangleInscribed.jpg I really think that the poster should post the actual diagram..this problem isnt as hard if the diagram is given... Ok lets see.. side of Square=S...side of Equilatral=A.. suppose point B and C are X..(btw you will quickly see that B and C are exactly X away)..i.e BE=CE.. so A^2=S^2+(sx)^2 similary BC=A => A^2=x^2+x^2 or 2x^2 area of ADB=1/2 (S)*(sx) area of BCE=1/2 (X)*(X) Ok, so we S^2  2xS + x^2=2X^2 x^2=2s^22xs or 2(s^2xs) area of ADB =1/2* (s^2xs) area of BCE=1/2*x^2 we know x^2 interms of S.. ADB/BCE = (s^2xs)/2(s^2xs)=1 ratio of BCE to ADB=2 took about 2 mins



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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Updated on: 17 Jul 2008, 19:06
jallenmorris wrote: durgesh, can you label which of these goes to which triangle and how you came up with \(\frac{\sqrt{3}}{4}\)?
\(\frac{\sqrt{3}}{4}\) * 2a^2
1/2 a^2
2 * 1/2 * b(a+b) \(\frac{\sqrt{3}}{4}\) * 2a^2  Equilateral triangle ABC with side \(a * sqrt2\) 1/2 a^2  trinagle BCE 2 * 1/2 * b(a+b)  traingle ADB + traingle ACF
Originally posted by durgesh79 on 17 Jul 2008, 19:00.
Last edited by durgesh79 on 17 Jul 2008, 19:06, edited 1 time in total.



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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16 Mar 2015, 05:04
jallenmorris wrote: Here is how I approached the problem. AD (side of square) = 1 BE (side of 45:45:90 triangle) = y CB (side of equilateral triangle) = z We're looking for the ratio of the area of BEC to ADB, or \(\frac{BEC}{ADB}\) If area of Tri. BEC = \(\frac{y^2}{2}\) and Tri.ADB = \(\frac{1(1y)}{2}\) then we have \(\frac{y^2}{2} * \frac{2}{1y} = \frac{y^2}{1y}\) if y = \(\frac{1}{2}\), then y^2 = 1/4 and 1y = 1/2, so we get 1/4 : 1/2 or 1/2....that's not an option. Attachment: TriangleInscribed.jpg We can't take any value for y. There's only one way to put an equilateral triangle inside a square if we fix the square side, so if we chose that side then y is not flexible, and if you try for y=1/3 you'll not find the correct ratio for example. I have a solution: Let's take a=1 as the side of the square, and let's name BE=x. Area of ADB = 1.(1x)/2=(1x)/2 Area of BEC=x^2/2 Therefore, ratio = x^2/(1x) Let's find x: AE=sqrt(2) Side of the Equilateral Triangle = x*sqrt(2) because BEC is isosceles. On the other hand, if we apply Pythagoras on triangle ADB: (1x)^2 = (x*sqrt(2))^2 1 = 2x^21 which leads to a quadratic equation with one positive solution: x=sqrt(3)  1 Let's then replace x with its value in ratio = x^2/(1x) > ratio= 2



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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25 Apr 2016, 07:14
just skip the question at the real exam! it takes a while just to figure out how to place the triangle into the square and then the whole work is about algebraic manipulation involving quadtatics. and most important  picking smart numbers just won't work here. a great question to waste >3 min and mental energy and get it wrong



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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01 May 2016, 06:37
tarek99 wrote: An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles: ADB, ACF and BEC. What is the ratio of the area of triangle BEC to that of triangle ADB?
A. 4/3 B. \(sqrt(3)\) C. 2 D. 5/2 E. \(sqrt(5)\) Hi Bunuel, VeritasPrepKarishma, chetan2u, Can you please provide some easy explanation for this question
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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01 May 2016, 22:39
smartguy595 wrote: tarek99 wrote: An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles: ADB, ACF and BEC. What is the ratio of the area of triangle BEC to that of triangle ADB?
A. 4/3 B. \(sqrt(3)\) C. 2 D. 5/2 E. \(sqrt(5)\) Hi Bunuel, VeritasPrepKarishma, chetan2u, Can you please provide some easy explanation for this question Hi, Here is a solution for the Q.. See the attached figure let the sides be 4 and BE be x, than BD = 4x.. Now we have made a equilateral triangle .. so its side from triangle BCE = \(\sqrt{x^2+x^2}\) = \(\sqrt{2}x\).. Now look at the right angle triangle ABD.. here AB^2 = AD^2 + BD^2.. \((\sqrt{2}x)^2 = 4^2 + (4x)^2\)... \(2x^2 = 16 + 16 + x^2  8x..\) \(x^2 + 8x  32 = 0\).. Roots of Quad eq ax^2+bx+c=0 are \(b+ \sqrt{b^24ac}/2\).. the VALID value of x comes out as \(4(\sqrt{3}1)\).. Now area of ABD = \(\frac{1}{2} * 4 * (4x) = 2*(4  4(\sqrt{3}1))\) .. => \(2*(84\sqrt{3}) = 8(2\sqrt{3})\).. Area of BCE = \(\frac{1}{2} *x*x = \frac{1}{2} (4(\sqrt{3}1))^2 = \frac{1}{2}*16*(\sqrt{3}1)^2 = 8(3+12\sqrt{3})\).. =>\(8(42\sqrt{3})= 16(2\sqrt{3})\).. so the ratio of areas of \(BCE/ABD = 16(2\sqrt{3})/8(2\sqrt{3})= 2\) C Ofcourse there is a geometrical approach too.. Try it
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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17 Aug 2018, 16:45
Hi Bunuel! Could you plz explain how to establish that triangles ACF and ADB are congruent? We have two sides(AF=AD & AC=AB) and one nonincluded angle (angles AFC & ADB) same in both the triangles. I understand that it is none of the cases from SSS,SAS,ASA or AAS.
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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17 Sep 2019, 17:52
Very tricky question. The actual setup of the diagram took a bit of time to figure out.
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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08 Oct 2019, 04:53
tarek99 wrote: An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles: ADB, ACF and BEC. What is the ratio of the area of triangle BEC to that of triangle ADB?
A. 4/3
B. \(\sqrt 3\)
C. 2
D. 5/2
E. \(\sqrt 5\) jallenmorris wrote: Here is how I approached the problem. AD (side of square) = 1 BE (side of 45:45:90 triangle) = y CB (side of equilateral triangle) = z We're looking for the ratio of the area of BEC to ADB, or \(\frac{BEC}{ADB}\) If area of Tri. BEC = \(\frac{y^2}{2}\) and Tri.ADB = \(\frac{1(1y)}{2}\) then we have \(\frac{y^2}{2} * \frac{2}{1y} = \frac{y^2}{1y}\) if y = \(\frac{1}{2}\), then y^2 = 1/4 and 1y = 1/2, so we get 1/4 : 1/2 or 1/2....that's not an option. Attachment: TriangleInscribed.jpg Area of BEC = y^2/2 Area of ADB = 1(1y)/2 Need = Area of BEC/Area of ADB = y^2/(1y) AB = BC (Sides of equilateral triangle) 1 + 1 + y^2  2y = 2y^2 2 + 2y = y^2 2 (1  y) = y^2 y^2/(1y) = 2 ANSWER: C



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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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27 Dec 2019, 08:35
kevincan wrote: Since AB=BC,\(x^2 + 2xy + y^2 +x^2 = 2y^2, so 2x^2 +2xy =2x(x+y)= y^2\) The ratio of the area of triangle BEC to that of triangle ADB is \(y^2/x(x+y) =2\) Why is DB = CF ?




Re: An equilateral triangle ABC is inscribed in square ADEF, forming three
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