If the rectangular faces of a brick have their diagonals in the ratio 3 : 2\sqrt3 : \sqrt15, then the ratio of the length of the shortest edge of the brick to that of its longest edge is

CAT 2019 Shift 1 Paper

Question : 74 of 100

Marks: +1, -0

If the rectangular faces of a brick have their diagonals in the ratio

3 : 2√3 : √15

, then the ratio of the length of the shortest edge of the brick to that of its longest edge is

Solution:

Let the edges of the brick be

a, b

, and

c

such that $a

a^{2}+b^{2}=3^{2}=9 . . . .(1)

a^{2}+c^{2}=(2 √{3})^{2}=12 . . .(2)

b^{2}+c^{2}=(√{15})^{2}=15 . . .(3)

Adding all three equations. We get

2(a^{2}+b^{2}+c^{2})=9+12+15=36

a^{2}+b^{2}+c^{2}=18 . . .(4)

From (1) and

(4), c=3

From (3) and

(4), a=√{3}

Therefore, required ratio

=\;{a}/{c}=\;{√{3}}/{3}=\;{1}/{√{3}}

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