REASONING Triangles $$ A B C $$ and $$ D E F $$ are similar. Which statement is correct? Select all that apply. (A) $$ \frac{B C}{E F}=\frac{A C}{D F} $$ (B) $$ \frac{A B}{D E}=\frac{C A}{F E} $$ (C) $$ \frac{A B}{E F}=\frac{B C}{D E} $$ (D) $$ \frac{C A}{F D}=\frac{B C}{E F} $$ 7. USING STRUCTURE Rectangle $$ A $$ is similar to rectangle $$ B $$. Rectangle $$ A $$ has side lengths of 6 anc 12. Rectangle $$ B $$ has a side length of 18 . What are the possible values for the length of the other side rectangle B? Select all that apply. (A) 6 (B) 9 (C) 24 (D) 36

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To solve the problems, we will analyze each part step by step.

### Part 1: Similar Triangles

Given that triangles $$ A B C $$ and $$ D E F $$ are similar, we know that the ratios of the lengths of corresponding sides are equal. This means that for any corresponding sides $$ a, b, c $$ of triangle $$ A B C $$ and $$ d, e, f $$ of triangle $$ D E F $$, the following relationships hold:

$$
\frac{a}{d} = \frac{b}{e} = \frac{c}{f}
$$

Now, let's evaluate each statement:

**(A)** $$ \frac{B C}{E F}=\frac{A C}{D F} $$

This statement is true because it follows the property of similar triangles where the ratio of one pair of corresponding sides equals the ratio of another pair.

**(B)** $$ \frac{A B}{D E}=\frac{C A}{F E} $$

This statement is also true. It follows the same reasoning as (A) since $$ A B $$ corresponds to $$ D E $$ and $$ C A $$ corresponds to $$ F E $$.

**(C)** $$ \frac{A B}{E F}=\frac{B C}{D E} $$

This statement is false. The correct relationship should involve the ratios of corresponding sides, but $$ E F $$ does not correspond to $$ A B $$ in the same triangle.

**(D)** $$ \frac{C A}{F D}=\frac{B C}{E F} $$

This statement is false. Similar triangles maintain the ratio of corresponding sides, but $$ F D $$ does not correspond to $$ C A $$.

### Summary for Part 1:
The correct statements are:
- (A)
- (B)

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### Part 2: Similar Rectangles

Given that rectangle $$ A $$ is similar to rectangle $$ B $$ and has side lengths of 6 and 12, we can find the scale factor between the rectangles. The side lengths of rectangle $$ B $$ are given as one side length of 18.

The ratio of the sides of rectangle $$ A $$ is:

$$
\text{Ratio} = \frac{6}{12} = \frac{1}{2}
$$

Now, if one side of rectangle $$ B $$ is 18, we can find the other side using the scale factor. Let $$ x $$ be the length of the other side of rectangle $$ B $$:

$$
\frac{6}{x} = \frac{1}{2} \implies 6 \cdot 2 = 1 \cdot x \implies x = 12
$$

Now, we can also find the other side using the ratio of the longer side:

$$
\frac{12}{x} = \frac{1}{2} \implies 12 \cdot 2 = 1 \cdot x \implies x = 24
$$

Thus, the possible values for the other side of rectangle $$ B $$ can be calculated as follows:

1. If the longer side is 18, the shorter side must be $$ 9 $$ (since $$ \frac{6}{x} = \frac{12}{18} $$).
2. If the shorter side is 18, the longer side must be $$ 36 $$ (since $$ \frac{12}{x} = \frac{6}{18} $$).

### Summary for Part 2:
The possible values for the length of the other side of rectangle $$ B $$ are:
- (B) 9
- (D) 36

### Final Answers:
1. For similar triangles: (A), (B)
2. For similar rectangles: (B), (D)
For similar triangles, the correct statements are (A) and (B). For similar rectangles, the possible lengths for the other side are (B) 9 and (D) 36.

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