Question

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

• A. √3 : 2
• B. 2 : √5
• C. 1 : √3
• D. √2 : √3

Answer 1

The Correct Option is C

Let the edges of the brick be \( a, b, c \) such that \( a < b < c \).

We are given:

• \( a^2 + b^2 = 9 \quad \text{(1)} \)
• \( a^2 + c^2 = (2\sqrt{3})^2 = 12 \quad \text{(2)} \)
• \( b^2 + c^2 = (\sqrt{15})^2 = 15 \quad \text{(3)} \)

Adding all three equations: \[ (a^2 + b^2) + (a^2 + c^2) + (b^2 + c^2) = 9 + 12 + 15 = 36 \] \[ \Rightarrow 2(a^2 + b^2 + c^2) = 36 \Rightarrow a^2 + b^2 + c^2 = 18 \quad \text{(4)} \]

Now, subtracting Equation (1) from Equation (4): \[ c^2 = 18 - (a^2 + b^2) = 18 - 9 = 9 \Rightarrow c = 3 \]

Similarly, subtracting Equation (3) from Equation (4): \[ a^2 = 18 - (b^2 + c^2) = 18 - 15 = 3 \Rightarrow a = \sqrt{3} \]

Hence, the required ratio is: \[ \frac{a}{c} = \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \]

Final Answer: \( \boxed{\frac{1}{\sqrt{3}}} \)