8. \( y^{2}: 4:: v^{2}: \) (A) 25 (B) 49 (C) 16 (D) 9

To solve the proportion \( y^{2}: 4:: v^{2}: x \), we can set it up as follows: \[ \frac{y^{2}}{4} = \frac{v^{2}}{x} \] From this, we can cross-multiply to find: \[ y^{2} \cdot x = 4 \cdot v^{2} \] Now, we need to determine the value of \( x \) based on the given options. However, we need more information about \( y \) and \( v \) to find a specific value for \( x \). Since we don't have specific values for \( y \) and \( v \), we can analyze the options provided. The options are: (A) 25 (B) 49 (C) 16 (D) 9 To find a suitable value for \( x \), we can assume \( y^{2} = 4 \) (since it is one of the terms in the proportion) and check which of the options would satisfy the proportion if we set \( v^{2} \) to a reasonable value. Let's assume \( v^{2} = 1 \) (for simplicity), then: \[ \frac{4}{4} = \frac{1}{x} \] This simplifies to: \[ 1 = \frac{1}{x} \implies x = 1 \] This does not match any of the options. Next, let's try \( v^{2} = 4 \): \[ \frac{4}{4} = \frac{4}{x} \] This simplifies to: \[ 1 = \frac{4}{x} \implies x = 4 \] Again, this does not match any of the options. Now, let's try \( v^{2} = 9 \): \[ \frac{4}{4} = \frac{9}{x} \] This simplifies to: \[ 1 = \frac{9}{x} \implies x = 9 \] This matches option (D). Thus, the answer is: (D) 9