https://file.aiquickdraw.com/mathbot/file/1730132077285jskzpuqz.jpg

Let’s break this problem down step-by-step:

### **Given:**
- Two rectangles:  
  - The first rectangle (ABCD) has side lengths of 4 (horizontal) and 2 (vertical).
  - The second rectangle (EFGH) has side lengths of 6 (horizontal) and 4 (vertical).

### **Statements to Analyze:**
1. **Jada’s Argument:**  
   - "These rectangles are similar because all the side lengths differ by 2."

2. **Lin’s Argument:**  
   - "These rectangles are similar because I can dilate the sides of rectangle ABCD using different scale factors:  
     - Scale the **vertical sides (AD and BC)** by a factor of **2** (from 2 to 4).  
     - Scale the **horizontal sides (AB and CD)** by a factor of **1.5** (from 4 to 6).
     - Then use a translation to align the rectangles."

### **Reasoning:**

#### **What Does Similarity Mean?**
- Two figures are similar if their corresponding angles are equal and their side lengths are proportional by the **same scale factor**.

#### **Analyzing Jada's Argument:**
- Jada says that the rectangles are similar because the side lengths differ by 2 units (6 - 4 = 2 and 4 - 2 = 2).
- **Issue with Jada’s Reasoning:**  
  Similarity depends on **proportions, not the difference in side lengths**. So, her argument is incorrect. For similarity, both rectangles need to have a consistent scale factor between their corresponding sides, not just a constant difference.

#### **Analyzing Lin’s Argument:**
- Lin suggests using different scale factors for the horizontal and vertical sides:
  - Horizontal sides: \( \frac{6}{4} = 1.5 \)
  - Vertical sides: \( \frac{4}{2} = 2 \)
- **Issue with Lin’s Reasoning:**  
  For two figures to be similar, **all corresponding side lengths must share the same scale factor**. However, Lin is using two different scale factors (1.5 and 2), which means the rectangles are **not similar**. Lin’s reasoning is also incorrect.

### **Conclusion:**
- **Neither Jada nor Lin** is correct.  
- The two rectangles are **not similar** because the ratio between their corresponding sides is not the same. In other words, there is no single scale factor that can transform one rectangle into the other.

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Do you want more details or have any questions?

### 5 Related Questions:
1. What are the conditions for two figures to be congruent?
2. How do you find the scale factor between two similar shapes?
3. Can two shapes with different proportions ever be congruent?
4. How do translations and dilations differ in transformations?
5. Can two rectangles ever be similar if only one pair of sides is proportional?

### Tip:
When analyzing similarity, always check if **all corresponding sides** have the **same ratio** — it's not enough to compare the absolute differences.

Jada says, “These rectangles are similar because all of the side lengths differ by 2.” Lin says, “These rectangles are similar. I can dilate AD and BC using a scale factor of 2 and AB and CD using a scale factor of 1.5 to make the rectangles congruent. Then I can use a translation to line up the rectangles.” Do you agree with either Jada or Lin? Explain your reasoning.

Explore a problem on similarity in rectangles as Jada and Lin present their arguments. This geometry problem involves analyzing similarity and transformations, and understanding scale factors in rectangular shapes. Ideal for Grades 7-9.

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