https://file.aiquickdraw.com/mathbot/file/1726705412680srqa11v5.jpg

In the image, we are given two rectangles \( ABCD \) and \( EFGH \), which are similar (denoted by \( ABCD \sim EFGH \)). This means that the corresponding sides of these rectangles are proportional.

### Step-by-step approach:

1. **Dimensions of rectangle ABCD:**
   - The sides are labeled as \( AB = 2 \, \text{cm} \), \( BC = 3 \, \text{cm} \), and so on. 
   
2. **Dimensions of rectangle EFGH:**
   - The corresponding sides to those of \( ABCD \) are labeled as \( EH = 8 \, \text{cm} \) and \( FG = k \, \text{cm} \).

3. **Proportions:**
   Since the rectangles are similar, the ratio of corresponding sides must be equal:
   \[
   \frac{EH}{AB} = \frac{FG}{BC}
   \]
   Plugging in the known values:
   \[
   \frac{8}{2} = \frac{k}{3}
   \]
   
4. **Solving for \( k \):**
   Cross-multiply to solve for \( k \):
   \[
   8 \times 3 = 2 \times k
   \]
   \[
   24 = 2k
   \]
   \[
   k = \frac{24}{2} = 12 \, \text{cm}
   \]

Therefore, the value of \( k \) is **12 cm**.

Would you like further details or have any other questions?

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Here are 5 related questions you might find useful:
1. What is the definition of similar polygons?
2. How do the properties of proportions apply to solving for unknown sides in similar shapes?
3. Can the same proportional reasoning be used for non-rectangular shapes?
4. How would you apply this concept if the rectangles were not aligned with the coordinate plane?
5. What happens to the area of similar shapes when scaling up or down?

**Tip:** When working with similar figures, always remember that all corresponding sides must have the same ratio, but angles remain equal.

ABCD ~ EFGH. What is the value of k? Provide an answer accurate to the nearest tenth.

This problem involves two similar rectangles ABCD and EFGH. Using the properties of proportionality between corresponding sides, you will learn how to solve for the unknown value k. Given that ABCD ~ EFGH, the solution involves setting up a proportion and solving for k. The sides of the rectangles are provided, and the solution is calculated to be 12 cm.

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