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# If the area of an equilateral triangle with side t is equal

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Manager
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If the area of an equilateral triangle with side t is equal [#permalink]  05 Sep 2007, 12:24
If the area of an equilateral triangle with side t is equal to the area of a square with side s, what is the ratio of t to s?

a) 2:3
b) 16:3
c) 4:(3^1/2)
d) 2:(3^1/4)
e) 4:(3^1/4)
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Re: GMATPrep; geometry [#permalink]  05 Sep 2007, 12:58
gnr646 wrote:
If the area of an equilateral triangle with side t is equal to the area of a square with side s, what is the ratio of t to s?

a) 2:3
b) 16:3
c) 4:(3^1/2)
d) 2:(3^1/4)
e) 4:(3^1/4)

D.

(1/2) * t * t*sqrt(3)/2 = s^2
(t/s)^2 = 4/sqrt(3)
t/s = 2 / 3^1/4
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Re: GMATPrep; geometry [#permalink]  05 Sep 2007, 22:12
gnr646 wrote:
If the area of an equilateral triangle with side t is equal to the area of a square with side s, what is the ratio of t to s?

a) 2:3
b) 16:3
c) 4:(3^1/2)
d) 2:(3^1/4)
e) 4:(3^1/4)
I got C.

a) t^2*(sqrt)3/4 = s^2

b) (1/s)t^2*((sqrt)3)/4 = s^2(1/s)

c) t^2*((sqrt)3)/4*(1/s) = s

d) t^2*((sqrt)3)/4s = s

e) t/s = t/t^2*((sqrt)3)/4s = t*4s/t^2*((sqrt)3) = 4s/t((sqrt)) = 4s/((t)(3^1/2))

What is the OE?
Director
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Re: GMATPrep; geometry [#permalink]  05 Sep 2007, 22:19
gnr646 wrote:
If the area of an equilateral triangle with side t is equal to the area of a square with side s, what is the ratio of t to s?

a) 2:3
b) 16:3
c) 4:(3^1/2)
d) 2:(3^1/4)
e) 4:(3^1/4)

area of the square = s^2
area of the equilateral triangle = (sqrt3/4) (t^2)

area of the square = area of the equilateral triangle
s^2 = (sqrt3/4) (t^2)
s = (3^(1/4) /2) (t)
t/s = 2/3^(1/4)

D.
Manager
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In an equirateral triangle of side t, let x be the height. Then, by Pythagoras theorem, t^2 = (t/2)^2 + x^2 hence x = tsqrt(3)/2. The area of a triangle is 1/2(height)*base = 1/2*t*sqrt(3)*t/2 = t^2*sqrt(3)/4 (1) This is equal to s^2. Thefore,

t^2 * sqrt(3) = 4*s^2 and hence, t^2/s^2 = 4/sqrt(3) therefore t/s = 2*(1/3^(1/4))
GMAT Club Legend
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Area of equilateral triangle = t^2(sqrt3)/4
Area of square = s^2

So t^2(sqrt3) = 4s^2

t^2/s^2 = 4/sqrt3 = 4/3^1/2

t/s = 2/3^1/4 --> Ans = D
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Re: GMATPrep; geometry [#permalink]  06 Sep 2007, 20:28
gnr646 wrote:
If the area of an equilateral triangle with side t is equal to the area of a square with side s, what is the ratio of t to s?

a) 2:3
b) 16:3
c) 4:(3^1/2)
d) 2:(3^1/4)
e) 4:(3^1/4)

I get D.

I first tried numbers but this wasnt getting me anywhere.

So I just used variables. A of T=b*h/2. A of S: S1*S2.

in this case A of the triangle is: 1/2t*1/2tsqrt3. The reason its 1/2sqrt3 is because when u split a equilateral into two it becomes to 30-60-90 triangles. the base corresponds to angle 30 and is 1/2t. So the height is 1/2tsqrt3.

A of the square is just s*s=s^2.

make the equations equal to eachother. s^2=1/2t*1/2tsqrt3. --> s^2=1/4t^2sqrt3. Sqrt both sides. --> s=1/2t *3^1/4.

t/s=1/2*3^1/4---> 2/3^1/4.
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Clearly, D

Easy one
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