If the two regions above have the same area, what is the ratio of t:s?

Question

https://gmatclub.com/forum/download/file.php?id=20492 If the two regions above have the same area, what is the ratio of t:s? A. 2 : 3 B. 16 : 3 C. 4 : \sqrt{3} D. 2 : \sqrt {3} E. 4 : \sqrt {3} Triangle and square.gif 27.JPG

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=20492 
 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 {3}} .

Answer: D.

robertrdzak · Answer

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)

bhushan252 · Answer

Sqr (S) = Sqrt (3) / 4 * sqr (T)
on simplification T/S = 2 : (3)^(1/4)
 OA D

jpr200012 · Answer

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.

jpr200012 · Answer

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!

YourDreamTheater · Answer

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!

jpr200012 · Answer

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?

KarishmaB · Answer

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

spyguy · Answer

one quick question where I am stumped. When you square root a square root is that where you are getting the 4th root?

KarishmaB · Answer

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.

mejia401 · Answer

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

CEdward · Answer

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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If the two regions above have the same area, what is the ratio of t:s?

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If the two regions above have the same area, what is the ratio of t:s?

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