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Ratio of the side of the square to the side of the triangle.

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Ratio of the side of the square to the side of the triangle.

by Vincen » Fri Sep 08, 2017 2:17 pm
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) ∜3 : 4
(E) ∜3 : 2

OA is E.

Does it depends on the dimensions of the sides? or it is general? How can I find the ratio?
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by GMATGuruNY » Fri Sep 08, 2017 2:37 pm
Vincen 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) √3 : 4
(D) ∜3 : 4
(E) ∜3 : 2
Area of an equilateral triangle = (s²√3)/4.

Let the square have a side of 1, with the result that the area of the square = 1² = 1.

Since the equilateral triangle has the same area, we get:
(s²√3)/4 = 1
s² = 4/√3
s = 2/(∜3).

Resulting ratio:
(side of the square)/(side of the triangle) = 1/(2/(∜3)) = (∜3)/2.

The correct answer is E.
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by Jay@ManhattanReview » Fri Sep 08, 2017 10:06 pm
Vincen 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) √3 : 4
(D) ∜3 : 4
(E) ∜3 : 2

OA is E.

Does it depends on the dimensions of the sides? or it is general? How can I find the ratio?
Hi Vincen,

The answer to your question, "Does it depend on the dimensions of the sides?", the answer is No. For the given condition (Area of a square and area of an equilateral triangle are equal), the ratio of the side of the square to the side of the triangle would always be constant.

Let's see how.

Area of an equilateral triangle = a²√3/4; where a = side of an equilateral triangle

Area of a square = b²; where b = side of a square

Since area of a square = area of the equilateral triangle, we get,

b² = a²√3/4
b²/a² = √3 /4
b/a = ∜3/2

Thus,

Side of the square / Side of the triangle = ∜3/2.

The correct answer: E

Hope this helps!

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by Jeff@TargetTestPrep » Tue Jan 23, 2018 9:06 am
Vincen 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) √3 : 4
(D) ∜3 : 4
(E) ∜3 : 2
We can let x = the side of the triangle and y = the side of the square. Recall that if the base of an equilateral triangle is x, the height is (x/2)√3, and thus the area is ½ bh = (x^2)√3/4.

The area of the equilateral triangle is equal to the area of the square, so we have:

(x^2)√3/4 = y^2

(x^2)√3 = 4y^2

√3/4 = y^2/x^2

Taking the square root of both sides we have:

∜3/2 = y/x

Answer: E

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