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The diagonal of a square is the height in an equilateral triangle. If

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Bunuel
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The diagonal of a square is the height in an equilateral triangle. If [#permalink] New post   13 May 2021, 01:30
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The diagonal of a square is the height in an equilateral triangle. If the side of the square is x, what is the side of the equilateral triangle in terms of x?

A. √(2/3)*x

B. 2/√3*x

C. √3x

D. √(8/3)*x

E. 2√3*x



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D
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Re: The diagonal of a square is the height in an equilateral triangle. If [#permalink] New post   13 May 2021, 02:46
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Bunuel wrote:

The diagonal of a square is the height in an equilateral triangle. If the side of the square is x, what is the side of the equilateral triangle in terms of x?

A. √(2/3)*x

B. 2/√3*x

C. √3x

D. √(8/3)*x

E. 2√3*x



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\(\sqrt{2}\) * x = \(\frac{\sqrt{3}}{2}\) * a

a = side of equilateral triangle

a= 2\(\frac{\sqrt{2}}{\sqrt{3}} \)* x
=\(\sqrt{\frac{8}{3}}\) * x
D is the answer IMO
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Re: The diagonal of a square is the height in an equilateral triangle. If [#permalink] New post   13 May 2021, 03:03
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Bunuel wrote:

The diagonal of a square is the height in an equilateral triangle. If the side of the square is x, what is the side of the equilateral triangle in terms of x?

A. √(2/3)*x

B. 2/√3*x

C. √3x

D. √(8/3)*x

E. 2√3*x


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Side of square = x
So, diagonal =\( x \sqrt{2}\)
Also, height of an equilateral triangle = \(\frac{\sqrt{3}}{2} * a\)
So, \(\frac{\sqrt{3}}{2} * a = x\sqrt{2}\)
\(a = \frac{x * 2 * \sqrt{2}}{\sqrt{3}}\)
\(a = \frac{\sqrt{8}}{\sqrt{3}}x\) --(D)
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TarunKumar1234
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Re: The diagonal of a square is the height in an equilateral triangle. If [#permalink] New post   13 May 2021, 06:40
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The diagonal of a square is the height in an equilateral triangle. If the side of the square is x, what is the side of the equilateral triangle in terms of x?

Diagonal of square = \(\sqrt{2}\) x = Height of the equilateral triangle = \(\frac{\sqrt{3}}{2} *side\)

Or, side =\(\frac{ 2\sqrt{2}}{\sqrt{3}}*x\) =\( \sqrt{\frac{8}{3}}\) *x

So, I think D. :)
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Re: The diagonal of a square is the height in an equilateral triangle. If [#permalink] New post   13 May 2021, 08:46
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Length of square = x
Diagonal of square = √2 x

In the triangle we can se 30:60:90 forming.
Ratio of sides opposite to angles = y: √3 y: 2y
√3 y = √2x
y = √2x/ √3
angle opposite to 90, side of the equilateral triangle = 2y = 2√2x/√3

IMO D
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Re: The diagonal of a square is the height in an equilateral triangle. If [#permalink] New post   13 May 2021, 23:25
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Side of square = x
diagonal of square =x√2
also x√2 is the height of equilateral triangle
Let a be the side of triangle
So, a^2=(x√2)^2 + (a/2)^2

a=√(8/3)X

Ans D
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Re: The diagonal of a square is the height in an equilateral triangle. If [#permalink]
13 May 2021, 23:25
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