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# If the two regions above have the same area, what is the

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If the two regions above have the same area, what is the [#permalink]

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11 Oct 2009, 09:03
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If the two regions above have the same area, what is the ratio of t:s?

A. 2 : 3
B. 16 : 3
C. 4 : (3)^(1/2)
D. 2 : (3)^(1/4)
E. 4 : (3)^(1/4)
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 09:10, edited 1 time in total.
Renamed the topic and edited the question.
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11 Oct 2009, 09:25
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If the two regions above have the same area, what is the ratio of t:s?

A. 2 : 3
B. 16 : 3
C. 4 : (3)^(1/2)
D. 2 : (3)^(1/4)
E. 4 : (3)^(1/4)

Area of equilateral triangle is $$area_{equilateral}=t^2*\frac{\sqrt{3}}{4}$$;

Area of square is $$area_{square}=s^2$$;

As areas are equal, then $$t^2*\frac{\sqrt{3}}{4}=s^2$$ --> $$\frac{t^2}{s^2}=\frac{4}{\sqrt{3}}$$ --> $$\frac{t}{s}=\frac{2}{\sqrt[4]{3}}$$.

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Last edited by Bunuel on 11 Oct 2009, 14:59, edited 1 time in total.
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11 Oct 2009, 09:46
thanks, I was curious, how did you get 1/4 as part of the solution? When i was doing the problem I kept ending up with 2^(1/2) : 3^(1/4)
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14 Oct 2009, 04:27
Sqr (S) = Sqrt (3) / 4 * sqr (T)
on simplification T/S = 2 : (3)^(1/4)
OA D
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Re: Geometry problem - Equal areas between triangle and square [#permalink]

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15 Aug 2010, 22:31
A(triangle) = $$1/2 * t * (t/2)*sqrt{3} = (t^2sqrt{3}) / 4$$

A(square) = $$s^2$$

$$(t^2 sqrt{3}) / 4 = s^2$$ Areas are equal.

$$t^2 = 4s^2 / sqrt{3}$$ Isolate t.

$$t = sqrt{4s^2 / 3^{1/2}}$$ Take the square root of both sides.

$$t = sqrt{4s^2)} / sqrt{3^{1/2}}$$ Square root of a fraction: $$sqrt{a/b} = sqrt{a} / sqrt{b}$$

$$t = 2s / 3^{1/4}$$ Simplify.

$$t/s = 2 / 3^{1/4}$$ Finally, the ratio.

Last edited by jpr200012 on 15 Aug 2010, 22:38, edited 1 time in total.
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Re: Geometry problem - Equal areas between triangle and square [#permalink]

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15 Aug 2010, 22:32
I thought this was a good problem. I overlooked that the triangle was equilateral the first time. I was looking at the shape and not the labels. One reason to always redraw figures!
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Re: Geometry problem - Equal areas between triangle and square [#permalink]

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15 Aug 2010, 22:56
I find this problem to be really easy if you just plug in numbers.

Let's find the area of the triangle first, since finding the area of a square is easier to do with a given value.

Say t =2

Area of equilateral triangle with side of 2 = $$\sqrt{3}$$

Set this area equal to $$s^2$$ and take the square root of both sides

s = 3^(1/4)

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16 Aug 2010, 08:08
YourDreamTheater: That works really fast, too. I've been using plugging in numbers more lately for saving time.

Bunuel: How the heck do you keep track of all these topics? Can you add GMAT Prep tag to this topic?
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16 Aug 2010, 14:19
Simple question I got it wrong
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Re: PS - Same Area, Ratio? [#permalink]

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13 Dec 2010, 20:45
consultinghokie wrote:
(imagine a picture of an equilateral triangle with sides T and a square with sides S)

If the two regions above have the same area, what is the ratio of T:S?

2:3

16:3

4: sq root 3

2: fourth root 3

4: third root 3

Area of an equilateral triangle of side $$T = (\sqrt{3}/4)T^2$$

Area of square of side S = $$S^2$$

Given: $$(\sqrt{3}/4)T^2$$ = $$S^2$$

$$T^2/S^2 = 4/\sqrt{3}$$

$$T/S = 2/fourth root 3$$
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 05 Nov 2010 Posts: 63 Followers: 1 Kudos [?]: 1 [0], given: 5 Re: PS - Same Area, Ratio? [#permalink] ### Show Tags 13 Dec 2010, 21:18 one quick question where I am stumped. When you square root a square root is that where you are getting the 4th root? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7076 Location: Pune, India Followers: 2089 Kudos [?]: 13305 [1] , given: 222 Re: PS - Same Area, Ratio? [#permalink] ### Show Tags 13 Dec 2010, 21:26 1 This post received KUDOS Expert's post spyguy wrote: one quick question where I am stumped. When you square root a square root is that where you are getting the 4th root? Yes. $$\sqrt{3} = 3^{\frac{1}{2}}$$ When you take the root again, you get $$(3^{\frac{1}{2}})^{\frac{1}{2}}$$ which is equal to $$3^{\frac{1}{4}}$$ In other words, it the fourth root of 3. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: PS - Same Area, Ratio? [#permalink]

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13 Dec 2010, 21:31
Thank you very much Karishma. Kudos! +1!
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14 Dec 2010, 00:19
Merging similar topics.
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Re: If the two regions above have the same area, what is the [#permalink]

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05 Aug 2014, 22:27
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If the two regions above have the same area, what is the [#permalink]

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29 Jun 2015, 15:51
A. 2 : 3 formula of area for equilateral triangles includes irrational number and area of square is the sides squared, a result without irrational number. One side must have an irrational number and therefore 2:3 cannot not be correct.
B. 16 : 3 same reasoning as above.
C. 4 : (3)^(1/2) Trick to see whether the final root was taken
D. 2 : (3)^(1/4) True statement
E. 4 : (3)^(1/4) Trick to test whether you're precise enough when selecting answer choices.

IMO D
If the two regions above have the same area, what is the   [#permalink] 29 Jun 2015, 15:51
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