Brick Edge Ratios from Face Diagonals

CAT 2019 Slot 1 · QA · Medium · Mensuration

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If the rectangular faces of a brick have their diagonals in the ratio 3:23:153 : 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

  1. A.

    1 : \sqrt{3}

  2. B.

    2 : \sqrt{5}

  3. C.

    \sqrt{2} : \sqrt{3}

  4. D.

    \sqrt{3} : 2

Answer

A

Explanation

Let dimensions be a,b,ca, b, c. a2+b2=9ka^2 + b^2 = 9k, b2+c2=12kb^2 + c^2 = 12k, c2+a2=15kc^2 + a^2 = 15k. Sum: 2(a2+b2+c2)=36k  ⟹  a2+b2+c2=18k2(a^2 + b^2 + c^2) = 36k \implies a^2 + b^2 + c^2 = 18k. c2=18k−9k=9kc^2 = 18k - 9k = 9k, a2=18k−12k=6ka^2 = 18k - 12k = 6k, b2=18k−15k=3kb^2 = 18k - 15k = 3k. b2:a2:c2=3:6:9=1:2:3b^2 : a^2 : c^2 = 3 : 6 : 9 = 1 : 2 : 3. Shortest edge =b= b, longest edge =c= c. b:c=1:3=1:3b : c = \sqrt{1} : \sqrt{3} = 1 : \sqrt{3}.

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