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If the rectangular faces of a brick have their diagonals in the ratio 3 : 2 \sqrt{3} : \sqrt{15}, what is the ratio of the length of the shortest edge of the brick to that of its longest edge?

 Jun 17, 2020
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Let a, b, c be the edges.

\(\begin{cases}ab = 3\\bc = 2\sqrt 3\\ca = \sqrt{15}\end{cases}\)

 

(1) * (3) / (2) : 

\(a^2 = \dfrac{3\sqrt{15}}{2\sqrt 3} = \dfrac{3\sqrt 5}2\\ a = \sqrt{\dfrac{3\sqrt 5}2}\)

 

Similarly, 

\(b = \sqrt{\dfrac6{5}\sqrt5}\) and \(c = \sqrt{2\sqrt 5}\)

 

Required ratio = b : c = \(\sqrt 3 : \sqrt 5\)

 Jun 17, 2020

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