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605-655 Level|   Geometry|      
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Bunuel
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If an equilateral triangle and a square have the same area then what 02 Jan 2020, 14:32
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E
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If an equilateral triangle and a square have the same area then what is the ratio of the side of the square to the side of the triangle?


(A) \(1 : 2\)

(B) \(2 : 3\)

(C) \(\sqrt{3} : 4\)

(D) \(\sqrt[4]{3} : 4\)

(E) \(\sqrt[4]{3} : 2\)
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E
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If an equilateral triangle and a square have the same area then what 02 Jan 2020, 15:13
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area of equilateral triangle with side a = 0.25a^2✓3, while area of a square with side b = b^2

0.25a^2✓3 = b^2
0.25✓3 = (b/a)^2
(3)^0.25 / 2 = (b/a)

Final answer is (E)

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If an equilateral triangle and a square have the same area then what 03 Jan 2020, 06:33
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given
√3/4 * a^2 = s^2
√3^1/4 /2 = s/a
IMO E

If an equilateral triangle and a square have the same area then what is the ratio of the side of the square to the side of the triangle?


(A) 1:21:2

(B) 2:32:3

(C) 3‾√:43:4

(D) 3‾√4:434:4

(E) 3‾√4:2
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If an equilateral triangle and a square have the same area then what 03 Jan 2020, 11:31
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If an equilateral triangle and a square have the same area then what is the ratio of the side of the square to the side of the triangle?

(A) 1:2

(B) 2:3

(C) \(\sqrt{3}\):4

(D) \(\sqrt{√3}\):4

(E) \(\sqrt{√3}\):2

Let side of triangle = a, Area = \(\frac{√3}{4}a^2\)
Side of square = s, Area = \(s^2\)

Now, \(s^2\) = \(\frac{√3}{4}a^2\)
\(\frac{s}{a} = \frac{\sqrt{√3}}{4}\)

Answer D.
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If an equilateral triangle and a square have the same area then what 03 Jan 2020, 22:12
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If the l is the length of a side of the equilateral triangle and a is the length of a side of the square,
then Area of the equilateral triangle = 1/2 * l * h, and h=l * √3 / 2
hence Area of the triangle = l^2 * √3 /4 ------(1)
Area of the square = a^2 ----(2)
But the two areas are equal, hence (1)=(2)
l^2 * √3 /4 = a^2
so a = √((l^2 * √3 /4 )) = l * (3)^0.25 /2 -------(3)
a:l =( l * (3)^0.25 /2 )/ l = (3)^0.25 / 2

The answer is E.
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If an equilateral triangle and a square have the same area then what 04 Jan 2020, 03:21
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root (3)/ 4 a^2 = s^2
where a is side an equilateral triangle and s is side of square. Hence, s/a = option E
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If an equilateral triangle and a square have the same area then what 04 Jan 2020, 20:08
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Letting s = the length of the side of the square and t = the length of a side of the triangle, we can create the equation:

s^2 = (t^2√3)/4

4s^2 = t^2√3

s^2/t^2 = √3/4

s/t = ∜3/2

Answer: E
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If an equilateral triangle and a square have the same area then what 05 Jan 2020, 05:26
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Let side of equilateral triangle = a and side of the triangle = b
Area of equilateral triangle =√3/4 a^2
Area of the triangle = b^2
b^2 =√3/4 a^2
b =∜3/2 a
the ratio of the side of the square to the side of the triangle = b/a = (∜3/2 a)/a = ∜3/2 = ∜3 ∶2(E)
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If an equilateral triangle and a square have the same area then what 05 Jan 2020, 18:27
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The length of the edge of the equilateral triangle is: a
=> The area of the equilateral triangle is \(a * \frac{a\sqrt{3}}{2} : 2 = \frac{a^{2}\sqrt{3}}{4}\)
The length of the edge of the square is: b
=> The area of the square is: \(b^{2}\)

=> \(b^{2} =\frac{a^{2}\sqrt{3}}{4}\)
=> \(\frac{b}{a} = \frac{\sqrt[4]{3} }{ 2}\)
=> Choice E
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If an equilateral triangle and a square have the same area then what 05 Jan 2020, 19:56
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Quote:
If an equilateral triangle and a square have the same area then what is the ratio of the side of the square to the side of the triangle?

(A) 1:2
(B) 2:3
(C) √3:4
(D) 4‾√3:4
(E) 4‾√3:2

area_square:area_equilateral:
a^2=x^2√3/4, x^2=4a^2/√3, x=2a/3^(1/4)

side_square:side_equilateral
a:x, a:2a/3^(1/4), 3^(1/4):2

Ans (E)
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