To analyze this problem, let’s break down Jada and Lin's reasoning to see if either is correct.

Given Information

1. Rectangle ABCD:

   • Lengths of sides: $AB = 4$, $BC = 2$, $CD = 4$, $DA = 2$

2. Rectangle EFGH:

   • Lengths of sides: $EF = 6$, $FG = 4$, $GH = 6$, $HE = 4$

Claims

• Jada’s Claim: The rectangles are similar because each side length differs by 2 units.
• Lin’s Claim: The rectangles are similar. Lin suggests using dilation with scale factors on pairs of sides to make the rectangles congruent and then using translation.

Analysis

1. Similarity of Rectangles:

   • Definition of Similar Rectangles: Two rectangles are similar if their corresponding side lengths are proportional.

   • For rectangles $ABCD$ and $EFGH$, we need to check if the ratios of corresponding sides are the same.

   • Ratio of Corresponding Sides: $\frac{AB}{EF} = \frac{4}{6} = \frac{2}{3} \quad \text{and} \quad \frac{BC}{FG} = \frac{2}{4} = \frac{1}{2}$

   • Since $\frac{AB}{EF} \neq \frac{BC}{FG}$, the rectangles are not similar.

2. Evaluating Jada’s Reasoning:

   • Jada’s argument that the rectangles are similar because all side lengths differ by 2 units is incorrect. Similarity in geometry depends on the proportionality of corresponding side lengths, not on the difference between side lengths.

3. Evaluating Lin’s Reasoning:

   • Lin suggests dilating $AD$ and $BC$ by a scale factor of 2, and $AB$ and $CD$ by a scale factor of 1.5. However, similarity requires a consistent scale factor for all sides of the rectangle. Thus, using two different scale factors does not maintain similarity, which means Lin’s approach is also incorrect.

Conclusion

Both Jada and Lin are incorrect in claiming that the rectangles are similar.

Would you like a detailed breakdown of each step or have any questions?

Here are 5 follow-up questions based on this problem:

1. What is the formal definition of similar figures in geometry?
2. How do you determine if two rectangles are congruent instead of similar?
3. Why can’t we use different scale factors for different sides when establishing similarity?
4. Can two rectangles with different side lengths ever be similar? Why or why not?
5. What transformations maintain similarity in geometric figures?

Tip: Always ensure corresponding side lengths have the same ratio when checking for similarity in rectangles or other polygons.