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

We have a rectangular brick with three different rectangular faces. Each face has a diagonal and we're told these diagonals are in the ratio 3 : 2√3 : √15.

A rectangular brick has three dimensions (length width height). Let's call them a b and c.

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The three rectangular faces of our brick are:

Face 1: dimensions a × b → diagonal = $\sqrt{a^2 + b^2}$

Face 2: dimensions b × c → diagonal = $\sqrt{b^2 + c^2}$

Face 3: dimensions c × a → diagonal = $\sqrt{c^2 + a^2}$

For any rectangle with sides p and q the diagonal follows the Pythagorean theorem: diagonal = $\sqrt{p^2 + q^2}$.

Since the diagonals are in the ratio 3 : 2√3 : √15 we can write:

$\sqrt{a^2 + b^2} : \sqrt{b^2 + c^2} : \sqrt{c^2 + a^2} = 3 : 2\sqrt{3} : \sqrt{15}$

Let's introduce a constant k:

$\sqrt{a^2 + b^2} = 3k$

$\sqrt{b^2 + c^2} = 2\sqrt{3}k$

$\sqrt{c^2 + a^2} = \sqrt{15}k$

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$a^2 + b^2 = 9k^2$ ... (1)

$b^2 + c^2 = 12k^2$ ... (2)

$c^2 + a^2 = 15k^2$ ... (3)

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$(a^2 + b^2) + (b^2 + c^2) + (c^2 + a^2) = 9k^2 + 12k^2 + 15k^2$

$2a^2 + 2b^2 + 2c^2 = 36k^2$

$a^2 + b^2 + c^2 = 18k^2$ ... (4)

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Finding c:

From equations (1) and (4):

From (1): $a^2 + b^2 = 9k^2$

From (4): $a^2 + b^2 + c^2 = 18k^2$

Therefore: $c^2 = 18k^2 - 9k^2 = 9k^2$

So: $c = 3k$

Finding a:

From equations (2) and (4):

From (2): $b^2 + c^2 = 12k^2$

From (4): $a^2 + b^2 + c^2 = 18k^2$

Therefore: $a^2 = 18k^2 - 12k^2 = 6k^2$

So: $a = \sqrt{6}k$

Finding b:

From equations (3) and (4):

From (3): $c^2 + a^2 = 15k^2$

From (4): $a^2 + b^2 + c^2 = 18k^2$

Therefore: $b^2 = 18k^2 - 15k^2 = 3k^2$

So: $b = \sqrt{3}k$

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Our three dimensions are:

$a = \sqrt{6}k$ ≈ 2.45k

$b = \sqrt{3}k$ ≈ 1.73k

$c = 3k$ = 3k

Arranging in order: $b < a < c$

Shortest edge: $b = \sqrt{3}k$

Longest edge: $c = 3k$

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Ratio of shortest to longest edge = $b : c = \sqrt{3}k : 3k = \sqrt{3} : 3$

$\sqrt{3} : 3 = 1 : \sqrt{3}$

Therefore the ratio is $1 : \sqrt{3}$