Given the proportion: \( \frac{A}{B}=\frac{C}{D} \) Which of the following are equivalent proportions? I. \( \frac{D}{B}=\frac{C}{A} \) II. \( \frac{A}{C}=\frac{B}{D} \) III. \( \frac{A}{B}=\frac{D}{C} \) N. \( \frac{B}{A}=\frac{D}{C} \) III, N only I, II, III only I, II, N only None are equivalent

To determine which of the given options are equivalent to the proportion \( \frac{A}{B} = \frac{C}{D} \), let's analyze each one. 1. Transform the original proportion: From \( \frac{A}{B} = \frac{C}{D} \), we can cross-multiply to get \( A \cdot D = B \cdot C \). 2. Now check each option: - **I. \( \frac{D}{B} = \frac{C}{A} \)** - Cross-multiplying gives \( D \cdot A = B \cdot C \), which is equivalent and true. - **II. \( \frac{A}{C} = \frac{B}{D} \)** - Cross-multiplying gives \( A \cdot D = B \cdot C \), which is equivalent and true. - **III. \( \frac{A}{B} = \frac{D}{C} \)** - Cross-multiplying gives \( A \cdot C = B \cdot D \), which is the inverse and not equivalent. - **N. \( \frac{B}{A} = \frac{D}{C} \)** - This is just the reciprocal of III, which also does not hold since \( B \cdot C \neq A \cdot D \). So the options that are equivalent are I and II only. Thus, the correct answer is: I, II only.