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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
Sqr (S) = Sqrt (3) / 4 * sqr (T)
on simplification T/S = 2 : (3)^(1/4)
OA D
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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.

Originally posted by jpr200012 on 15 Aug 2010, 23:31.
Last edited by jpr200012 on 15 Aug 2010, 23:38, edited 1 time in total.
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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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)

Put t over s, and you have your answer!
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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consultinghokie
(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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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
one quick question where I am stumped. When you square root a square root is that where you are getting the 4th root?
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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spyguy
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.
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
So we have an equilateral triangle and a square.

A(triangle) = √3/4 x t^2
A(square) = s^2

(√3/4)t^2 / s^2 ---> t/s = 2 / 4√3

Answer is D.
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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Re: If the two regions above have the same area, what is the ratio of t:s? [#permalink]
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