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# If the ratio between the diagonal of a square and the height of an equ

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If the ratio between the diagonal of a square and the height of an equ  [#permalink]

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14 Sep 2018, 02:19
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65% (hard)

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60% (02:53) correct 40% (03:10) wrong based on 22 sessions

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If the ratio between the diagonal of a square and the height of an equilateral triangle is 5/3, respectively, what is the ratio of their areas?

A. $$\frac{5\sqrt{3}}{6}$$

B. $$\frac{5\sqrt{3}}{18}$$

C. $$\frac{25\sqrt{3}}{9}$$

D. $$\frac{25\sqrt{3}}{18}$$

E. 25/18

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Re: If the ratio between the diagonal of a square and the height of an equ  [#permalink]

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14 Sep 2018, 02:50
Bunuel wrote:
If the ratio between the diagonal of a square and the height of an equilateral triangle is 5/3, respectively, what is the ratio of their areas?

A. $$\frac{5\sqrt{3}}{6}$$

B. $$\frac{5\sqrt{3}}{18}$$

C. $$\frac{25\sqrt{3}}{9}$$

D. $$\frac{25\sqrt{3}}{18}$$

E. 25/18

ratio between the diagonal of a square and the height of an equilateral triangle is 5/3

$$(s√2) / (√3a/2) = 5/3$$
Here, s is the side of square and a is the side of equilateral triangle

$$s/a = 5√3 / 6√2$$

Ratio of areas = $$s^2 / [(√3/4)*a^2] = (4/√3)*(5√3 / 6√2)^2 = (4*75)/(72√3) = 25/6√3 =$$ $$\frac{25\sqrt{3}}{18}$$

Answer: option D
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Re: If the ratio between the diagonal of a square and the height of an equ  [#permalink]

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14 Sep 2018, 03:10
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Bunuel wrote:
If the ratio between the diagonal of a square and the height of an equilateral triangle is 5/3, respectively, what is the ratio of their areas?

A. $$\frac{5\sqrt{3}}{6}$$

B. $$\frac{5\sqrt{3}}{18}$$

C. $$\frac{25\sqrt{3}}{9}$$

D. $$\frac{25\sqrt{3}}{18}$$

E. 25/18

D/h = $$\frac{5}{3}$$
h = $$\sqrt{3}$$ a /2 where a is the side of the triangle.
D/a = $$\frac{5}{2\sqrt{3}}$$
Area of square = D^2/2
Area of equilateral triangle = $$\sqrt{3} a^2/4$$
Ratio = $$2(\frac{D}{a})^2 / \sqrt{3}$$ = $$\frac{25}{6{\sqrt{3}}}$$ = $$\frac{25\sqrt{3}}{18}$$

Answer D.
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Re: If the ratio between the diagonal of a square and the height of an equ   [#permalink] 14 Sep 2018, 03:10
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