Rectangle ABCD is similar to rectangle EFGH. The area of rectangle ABCD is 648 square inches, and the area of...

1. TRANSLATE the problem information

• Given information:
  • Rectangle ABCD is similar to rectangle EFGH
  • Area of ABCD = 648 square inches
  • Area of EFGH = 72 square inches
  • Longest side of ABCD = 36 inches
• Need to find: longest side of EFGH

2. INFER the key relationship

• Since the rectangles are similar, all corresponding sides are proportional by the same scale factor \(\mathrm{k}\)
• This means: if each side of EFGH is \(\mathrm{k}\) times the corresponding side of ABCD, then the area of EFGH is \(\mathrm{k^2}\) times the area of ABCD
• Why \(\mathrm{k^2}\)? Because area = length × width, so if both dimensions scale by \(\mathrm{k}\), area scales by \(\mathrm{k \times k = k^2}\)

3. TRANSLATE this relationship into an equation

• Area of EFGH = \(\mathrm{k^2}\) × Area of ABCD
• Substituting: \(\mathrm{72 = k^2 \times 648}\)

4. SIMPLIFY to find the scale factor

• \(\mathrm{k^2 = \frac{72}{648} = \frac{1}{9}}\)
• Take the square root: \(\mathrm{k = \sqrt{\frac{1}{9}} = \frac{1}{3}}\)
• This means each side of EFGH is \(\mathrm{\frac{1}{3}}\) the length of the corresponding side in ABCD

5. SIMPLIFY to find the longest side of EFGH

• Longest side of EFGH = \(\mathrm{k}\) × longest side of ABCD
• Longest side of EFGH = \(\mathrm{\frac{1}{3} \times 36 = 12}\) inches

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that areas scale by \(\mathrm{k^2}\) when sides scale by \(\mathrm{k}\). Instead, they assume areas scale by the same factor as the sides.

They might think: "Area of EFGH is 72 and area of ABCD is 648, so EFGH is \(\mathrm{\frac{72}{648} = \frac{1}{9}}\) the size. Therefore, each side of EFGH is \(\mathrm{\frac{1}{9}}\) the length of the corresponding side in ABCD."

This leads them to calculate: \(\mathrm{36 \times \frac{1}{9} = 4}\) inches, causing them to select Choice A (4).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly find \(\mathrm{k = \frac{1}{3}}\) but then get confused about which rectangle is larger and which scale factor to use.

They might calculate \(\mathrm{k^2 = \frac{72}{648} = \frac{1}{9}}\) correctly, but then think "ABCD is 9 times larger than EFGH in area" and incorrectly conclude the side ratio is \(\mathrm{9:1}\) instead of \(\mathrm{3:1}\). This leads them to select Choice B (9).

The Bottom Line:

The key insight that trips up most students is understanding that when similar figures have sides that differ by factor \(\mathrm{k}\), their areas differ by factor \(\mathrm{k^2}\). Missing this fundamental relationship about area scaling makes the entire problem unsolvable using the correct approach.