Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 6 times the length of the...

1. TRANSLATE the problem information

• Given information:
  • Rectangles ABCD and EFGH are similar
  • Each side of EFGH is 6 times the corresponding side of ABCD
  • Area of ABCD = 54 square units
  • Need to find: Area of EFGH

• What this tells us: Both the length AND width of EFGH are 6 times larger than the corresponding dimensions of ABCD.

2. INFER the area scaling relationship

• Key insight: When you scale up a rectangle's dimensions by a factor, the area doesn't scale by the same factor - it scales by the factor squared.

• Why? If original rectangle has length \(\mathrm{l}\) and width \(\mathrm{w}\), then area = \(\mathrm{l \times w}\)
• New rectangle has length \(\mathrm{6l}\) and width \(\mathrm{6w}\), so area = \(\mathrm{(6l) \times (6w)}\)
  \(\mathrm{= 36lw}\)
  \(\mathrm{= 36 \times (original\ area)}\)

• Therefore: Area of EFGH = \(\mathrm{36 \times Area\ of\ ABCD}\)

3. SIMPLIFY to find the final answer

• Area of EFGH = \(\mathrm{36 \times 54}\)
  \(\mathrm{= 1,944}\) square units

Answer: D. 1,944

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think that if the sides are 6 times larger, then the area is also 6 times larger. They miss the crucial insight that area scales by the square of the linear scale factor.

This incorrect reasoning: "Each side is 6 times bigger, so area is 6 times bigger: \(\mathrm{54 \times 6 = 324}\)"

This may lead them to select Choice C (324)

Second Most Common Error Path:

Poor TRANSLATE reasoning: Students misinterpret "each side of EFGH is 6 times the corresponding side of ABCD" and think this means the perimeter is 6 times larger, or get confused about what exactly is being scaled.

This creates confusion about what calculation to perform, leading to random guessing among the answer choices.

The Bottom Line:

The key challenge is recognizing that area scaling follows a quadratic relationship with linear scaling. Students who remember that "similar figures have areas that scale by \(\mathrm{k^2}\)" will solve this quickly, while those who assume linear scaling will consistently pick the wrong answer.