$$\begin{array}{c}6 \ 16 \ 4 \ 8\end{array}$$

Question

Examoogle · Accepted Answer

The correct answer is Option A: 6. In a proportion a : b :: c : d, the product of means (b × c) equals the product of extremes (a × d). Here, x × y = 6 × 6 = 36.

Step-by-Step Solution & Concepts:
Shortcut Trick: In a proportion a : b :: c : d, the product of means (b × c) equals the product of extremes (a × d). Here, x × y = 6 × 6 = 36.

Formula Used: $$x : 6 :: 6 : y \implies \frac{x}{6} = \frac{6}{y} \implies xy = 36$$

Given: Given ratio x : 6 :: 6 : y

Steps: Set up the proportion: The given proportion is $x : 6 :: 6 : y$. This means the ratio of $x$ to $6$ is equal to the ratio of $6$ to $y$.. Calculate the product xy: Cross-multiply to isolate the product of the terms: $x 	imes y = 6 	imes 6$.. Find the square root: The question asks for the value of $\sqrt{xy}$. Substituting $xy = 36$, we get $\sqrt{36} = 6$.

Answer: A is correct because the property of proportion dictates that $x 	imes y = 6 	imes 6 = 36$, so $\sqrt{xy} = \sqrt{36} = 6$.

Exam Tip: This concept of mean proportional is fundamental in geometry, specifically when calculating the altitude of a right-angled triangle relative to the segments of the hypotenuse.

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