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An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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38 00:00

Difficulty:   95% (hard)

Question Stats: 51% (02:53) correct 49% (03:05) wrong based on 292 sessions

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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. $$\sqrt 5$$

Originally posted by tarek99 on 17 Jul 2008, 07:11.
Last edited by Bunuel on 19 Jan 2018, 12:06, edited 3 times in total.
Added the OA.
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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7
2
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 $$y^2/x(x+y) =2$$
Attachments Dibujo12.jpg [ 10.28 KiB | Viewed 3762 times ]

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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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side of triangle BCE = a
side of square = (a+b)
two sides of triangle ACF are b and (a+b)

side of equilatrel triangle = $$a*sqrt(2)$$

sum of area of all triangles = area of square

sqrt(3)/4 * 2a^2 + 1/2 a^2 + 2 * 1/2 * b(a+b) = (a+b)^2

simplify this

b = (sqrt(3) - 1)/2 * a
a+b = a * ( sqrt(3) + 1) / 2

area of BCE = 1/2 a^2
area of AFC = 1/2 * b * (a+b)
= 1/2 * a^2 (3-1) / 4

ratio = 2: 1

answer C

time to solve : 7 minutes SVP  Joined: 30 Apr 2008
Posts: 1801
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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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 = $$\frac{1(1-y)}{2}$$ then we have
$$\frac{y^2}{2} * \frac{2}{1-y} = \frac{y^2}{1-y}$$

if y = $$\frac{1}{2}$$, then y^2 = 1/4 and 1-y = 1/2, so we get 1/4 : 1/2 or 1/2....that's not an option.

Attachment: TriangleInscribed.jpg [ 12.87 KiB | Viewed 9971 times ]

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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$. GMAT Club Premium Membership - big benefits and savings Senior Manager  Joined: 07 Jan 2008 Posts: 250 Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags tarek99 wrote: An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles: ADB, ACF and BEC. What is the ratio of the area of triangle BEC to that of triangle ADB? a) 4/3 b) sqrt(s) c) 2 d) 5/2 e) sqrt(5) I believe the size of these triangles can vary, meaning we can increase one to decrease the size of another. So, we can't come up with right answer unless they provide us pre-configured info. No? If not, I guess I would just press Next on this one VP  Joined: 21 Jul 2006 Posts: 1363 Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags durgesh79 wrote: side of triangle BCE = a side of square = (a+b) two sides of triangle ACF are b and (a+b) side of equilatrel triangle = $$a*sqrt(2)$$ sum of area of all triangles = area of square sqrt(3)/4 * 2a^2 + 1/2 a^2 + 2 * 1/2 * b(a+b) = (a+b)^2 simplify this b = (sqrt(3) - 1)/2 * a a+b = a * ( sqrt(3) + 1) / 2 area of BCE = 1/2 a^2 area of AFC = 1/2 * b * (a+b) = 1/2 * a^2 (3-1) / 4 ratio = 2: 1 answer C would you please explain how you got the side of the equilateral triangle to be time to solve : 7 minutes would you please explain how you got the side of the equilateral triangle to be $$a*sqrt(2)$$? SVP  Joined: 30 Apr 2008 Posts: 1801 Location: Oklahoma City Schools: Hard Knocks Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags if he makes 1 side, (the base) of BCE = a, it's a 45:45:90 triangle, so it has $$1:1:\sqrt{2}$$ sides. Therefore, one side of the triangle makes that hypotnuse and you have $$a\sqrt{2}$$ See the picture I drew above. I have the sides of BCE = y and the hypotnuse = z. He labeled his $$a$$ rather than y like I did. tarek99 wrote: durgesh79 wrote: side of triangle BCE = a side of square = (a+b) two sides of triangle ACF are b and (a+b) side of equilatrel triangle = $$a*sqrt(2)$$ sum of area of all triangles = area of square sqrt(3)/4 * 2a^2 + 1/2 a^2 + 2 * 1/2 * b(a+b) = (a+b)^2 simplify this b = (sqrt(3) - 1)/2 * a a+b = a * ( sqrt(3) + 1) / 2 area of BCE = 1/2 a^2 area of AFC = 1/2 * b * (a+b) = 1/2 * a^2 (3-1) / 4 ratio = 2: 1 answer C would you please explain how you got the side of the equilateral triangle to be time to solve : 7 minutes would you please explain how you got the side of the equilateral triangle to be $$a*sqrt(2)$$? _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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Originally posted by jallenmorris on 17 Jul 2008, 10:43.
Last edited by jallenmorris on 17 Jul 2008, 10:44, edited 1 time in total.
Senior Manager  Joined: 12 Jul 2008
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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tarek99 wrote:
durgesh79 wrote:
side of triangle BCE = a
side of square = (a+b)
two sides of triangle ACF are b and (a+b)

side of equilatrel triangle = $$a*sqrt(2)$$

sum of area of all triangles = area of square

sqrt(3)/4 * 2a^2 + 1/2 a^2 + 2 * 1/2 * b(a+b) = (a+b)^2

simplify this

b = (sqrt(3) - 1)/2 * a
a+b = a * ( sqrt(3) + 1) / 2

area of BCE = 1/2 a^2
area of AFC = 1/2 * b * (a+b)
= 1/2 * a^2 (3-1) / 4

ratio = 2: 1

answer C

would you please explain how you got the side of the equilateral triangle to be

time to solve : 7 minutes would you please explain how you got the side of the equilateral triangle to be $$a*sqrt(2)$$?

The sides are equal, so it's a 45/45/90 triangle. You know the sides are equal because the inscribed triangle is equilateral.
SVP  Joined: 30 Apr 2008
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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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)
_________________
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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a. GMAT Club Premium Membership - big benefits and savings Manager  Joined: 04 Apr 2008 Posts: 189 Location: Pune Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags How are you guys taking wo sides as contant i am not getting it...........as how the triangle is alligned can vary and so is the ratio of the sides....like allen has taken BE and CE as same is there any rule or was it just an asuumption _________________ Every Problem Has a Sloution So keep working AB VP  Joined: 21 Jul 2006 Posts: 1363 Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags jallenmorris wrote: if he makes 1 side, (the base) of BCE = a, it's a 45:45:90 triangle, so it has $$1:1:\sqrt{2}$$ sides. Therefore, one side of the triangle makes that hypotnuse and you have $$a\sqrt{2}$$ See the picture I drew above. I have the sides of BCE = y and the hypotnuse = z. He labeled his $$a$$ rather than y like I did. tarek99 wrote: durgesh79 wrote: side of triangle BCE = a side of square = (a+b) two sides of triangle ACF are b and (a+b) side of equilatrel triangle = $$a*sqrt(2)$$ sum of area of all triangles = area of square sqrt(3)/4 * 2a^2 + 1/2 a^2 + 2 * 1/2 * b(a+b) = (a+b)^2 simplify this b = (sqrt(3) - 1)/2 * a a+b = a * ( sqrt(3) + 1) / 2 area of BCE = 1/2 a^2 area of AFC = 1/2 * b * (a+b) = 1/2 * a^2 (3-1) / 4 ratio = 2: 1 answer C would you please explain how you got the side of the equilateral triangle to be time to solve : 7 minutes would you please explain how you got the side of the equilateral triangle to be $$a*sqrt(2)$$? but how can you decide that it's a 45:45:90 triangle? what if it's a 30:60:90 triangle? how can you be so sure which one it is? for example, if it were a 30:60:90 triangle, than the hypotenuse would be 2a, then what??? Originally posted by tarek99 on 17 Jul 2008, 10:54. Last edited by tarek99 on 17 Jul 2008, 11:02, edited 3 times in total. Senior Manager  Joined: 12 Jul 2008 Posts: 499 Schools: Wharton Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags apurva1985 wrote: How are you guys taking wo sides as contant i am not getting it...........as how the triangle is alligned can vary and so is the ratio of the sides....like allen has taken BE and CE as same is there any rule or was it just an asuumption There's only one way the triangle can align if it's an equilateral triangle. Manager  Joined: 04 Apr 2008 Posts: 189 Location: Pune Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags That is what i was trying to say is that the answer can vary from person to person..........Plz can somebody expalin it in detail......... _________________ Every Problem Has a Sloution So keep working AB SVP  Joined: 30 Apr 2008 Posts: 1801 Location: Oklahoma City Schools: Hard Knocks Re: An equilateral triangle ABC is inscribed in square ADEF, forming three [#permalink] ### Show Tags Because we know it is an equilateral triangle inscribe in a square and both the square and triangle share point A. So the sides coming from point A for the triangle towards the other sides fo the square are the same length. Look at my post above with the drawing in it. I believe that to be a rather accurate picture of this problem (although not drawn to scale or perfect angles). We know the inscribed triangle is an equialteral (also an iscoceles). In order for the 2 sides to extend out and hit the square, they must touch in the exact same spot (relative to the side) or one side of the triangle would be longer than the other and it would no longer be equilater (or iscoceles). tarek99 wrote: but how can you decide that it's a 45:45:90 triangle? what if it's a 30:60:90 triangle? how can you be so sure which one it is? _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a.

GMAT Club Premium Membership - big benefits and savings
Manager  Joined: 04 Apr 2008
Posts: 189
Location: Pune
Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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Even i was feeling that as if you try to distribute the angles what will be the angles of the triangle..........either what i am decepting is wrong or the problem is wrong..........i am going by the figure given by allen
jallenmorris wrote:
Because we know it is an equilateral triangle inscribe in a square and both the square and triangle share point A. So the sides coming from point A for the triangle towards the other sides fo the square are the same length.

Look at my post above with the drawing in it. I believe that to be a rather accurate picture of this problem (although not drawn to scale or perfect angles).

We know the inscribed triangle is an equialteral (also an iscoceles). In order for the 2 sides to extend out and hit the square, they must touch in the exact same spot (relative to the side) or one side of the triangle would be longer than the other and it would no longer be equilater (or iscoceles).

tarek99 wrote:

but how can you decide that it's a 45:45:90 triangle? what if it's a 30:60:90 triangle? how can you be so sure which one it is?

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VP  Joined: 21 Jul 2006
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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jallenmorris wrote:
Because we know it is an equilateral triangle inscribe in a square and both the square and triangle share point A. So the sides coming from point A for the triangle towards the other sides fo the square are the same length.

Look at my post above with the drawing in it. I believe that to be a rather accurate picture of this problem (although not drawn to scale or perfect angles).

We know the inscribed triangle is an equialteral (also an iscoceles). In order for the 2 sides to extend out and hit the square, they must touch in the exact same spot (relative to the side) or one side of the triangle would be longer than the other and it would no longer be equilater (or iscoceles).

tarek99 wrote:

but how can you decide that it's a 45:45:90 triangle? what if it's a 30:60:90 triangle? how can you be so sure which one it is?

fine, but this only tells me that ADB and AFC are equal and that DB is the same as CF. I still can't see how we choose the triangle to be either 45:45:90 or 30:60:90. I'm really sorry for troubling you, but would you please explain this point?
thanks
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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tarek99 wrote:
jallenmorris wrote:
Because we know it is an equilateral triangle inscribe in a square and both the square and triangle share point A. So the sides coming from point A for the triangle towards the other sides fo the square are the same length.

Look at my post above with the drawing in it. I believe that to be a rather accurate picture of this problem (although not drawn to scale or perfect angles).

We know the inscribed triangle is an equialteral (also an iscoceles). In order for the 2 sides to extend out and hit the square, they must touch in the exact same spot (relative to the side) or one side of the triangle would be longer than the other and it would no longer be equilater (or iscoceles).

tarek99 wrote:

but how can you decide that it's a 45:45:90 triangle? what if it's a 30:60:90 triangle? how can you be so sure which one it is?

fine, but this only tells me that ADB and AFC are equal and that DB is the same as CF. I still can't see how we choose the triangle to be either 45:45:90 or 30:60:90. I'm really sorry for troubling you, but would you please explain this point?
thanks

I think i have got it in (triangle ABD) angle BAD((90-60)/2)=15,angle ADB=90,angle ABD=75, using these values in triangle BCE angle cbe will be(180-75-60=45) so from here we got it
i suppose rest can be calculated with this
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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Attachment: TriangleExplanation.jpg [ 36.61 KiB | Viewed 9400 times ]

_________________
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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a.

GMAT Club Premium Membership - big benefits and savings
VP  Joined: 21 Jul 2006
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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perfect! that fills the gap in the puzzle! i'll now work through my calculations again to see whether i can get the final answer. thanks
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Re: An equilateral triangle ABC is inscribed in square ADEF, forming three  [#permalink]

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I really appreciate it. ok, what a brutal problem this has been! hehe....i managed to get to the correct answer. How about we all try to figure out whether there could be a faster way to solve this? I honestly don't believe that the GMAC guys actually expect us to use all these endless steps to arrive to our answer. There must be a much faster or more abstract way to solve this....hmm... Re: An equilateral triangle ABC is inscribed in square ADEF, forming three   [#permalink] 17 Jul 2008, 12:23

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

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