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# If an equilateral triangle and a square have the same area

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Manager
Joined: 13 Apr 2010
Posts: 90
If an equilateral triangle and a square have the same area [#permalink]

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05 Sep 2017, 12:13
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Difficulty:

45% (medium)

Question Stats:

61% (01:16) correct 39% (00:51) wrong based on 46 sessions

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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
Manager
Joined: 21 Feb 2017
Posts: 81
Re: If an equilateral triangle and a square have the same area [#permalink]

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05 Sep 2017, 12:17
E. ((Sqrt (3)*a^2/4)= b^2. Find b/a will be the answer.

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SC Moderator
Joined: 22 May 2016
Posts: 1765
If an equilateral triangle and a square have the same area [#permalink]

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05 Sep 2017, 13:27
sb0541 wrote:
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

Set the areas of each figure equal to find the ratio of sides.

Let S = side of square
Let s = side of triangle

Area of equilateral triangle:
$$\frac{s^2\sqrt{3}}{4}$$

Area of square: $$S^2$$

$$S^2= \frac{s{^2}\sqrt{3}}{4}$$ =

$$4(S^2) = {s^2\sqrt{3}}$$ =

$$\frac{S^2}{s^2} = \frac {\sqrt{3}}{4}$$

Take the square root of both sides:

$$\frac{S}{s} = \frac{\sqrt[4]{3}}{2}$$

For this problem, not knowing the formula for the area of an equilateral triangle would be difficult. Not impossible, even without trigonometry. Draw an equilateral triangle, drop an altitude, and use 30-60-90 triangle properties.
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If an equilateral triangle and a square have the same area   [#permalink] 05 Sep 2017, 13:27
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