In the figure above, rectangle ABCD is similar to rectangle EFGH, where vertices A, B, C, and D correspond to...

1. TRANSLATE the problem information

• Given facts:
  • Rectangle \(\text{ABCD} \sim \text{Rectangle EFGH}\) (similar rectangles)
  • Area of \(\text{ABCD} = 12\)
  • Side \(\mathrm{AD} = 2\)
  • Corresponding side \(\mathrm{EH} = 5\)
  • Need to find: Area of EFGH

• What "corresponding" means: Since the vertices correspond in order \(\mathrm{A \leftrightarrow E, B \leftrightarrow F, C \leftrightarrow G, D \leftrightarrow H}\), side AD on the first rectangle matches with side EH on the second rectangle.

2. INFER the solution strategy

• Key strategic insight: We have two similar rectangles and need to find how their areas relate. Since we know one side from each rectangle, we can:
  1. Find the scale factor between them
  2. Use the area scaling rule (areas scale by \(\mathrm{k}^2\), not k)

• Why this approach works: We don't need to find all dimensions—the area scaling rule gives us a direct path from one area to the other.

3. SIMPLIFY to find the scale factor

The scale factor k from rectangle ABCD to rectangle EFGH is:

\(\mathrm{k} = \frac{\mathrm{EH}}{\mathrm{AD}} = \frac{5}{2}\)

4. INFER the area relationship

Since the rectangles are similar with scale factor \(\mathrm{k} = \frac{5}{2}\):

• Linear measurements scale by k
• But areas scale by \(\mathrm{k}^2\) (this is crucial!)

Area ratio \(= \mathrm{k}^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}\)

5. SIMPLIFY to find the area of EFGH

Set up the proportion:

• \(\frac{\text{Area of EFGH}}{\text{Area of ABCD}} = \frac{25}{4}\)
• \(\frac{\text{Area of EFGH}}{12} = \frac{25}{4}\)
• \(\text{Area of EFGH} = 12 \times \frac{25}{4}\)

SIMPLIFY the arithmetic:

• \(\text{Area of EFGH} = \frac{12}{4} \times 25\)
• \(\text{Area of EFGH} = 3 \times 25\)
• \(\text{Area of EFGH} = 75\)

Answer: C. 75

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Using linear scale factor for area instead of squared scale factor

Many students correctly find \(\mathrm{k} = \frac{5}{2}\), but then incorrectly calculate:

• \(\text{Area of EFGH} = 12 \times \frac{5}{2} = 12 \times 2.5 = 30\)

This happens because they don't recognize (or forget) that while lengths scale by k, areas scale by \(\mathrm{k}^2\). They treat area as if it were a one-dimensional measurement.

This leads them to select Choice A (30).

Second Most Common Error:

Incomplete SIMPLIFY: Arithmetic error in computing \(12 \times \frac{25}{4}\)

Students might correctly set up \(\text{Area} = 12 \times \frac{25}{4}\) but then:

• Multiply \(12 \times 25 = 300\) first
• Forget to divide by 4, leaving their answer as 300
• Or divide incorrectly

However, since 300 is not among the answer choices, this error would cause confusion and likely lead to guessing or reworking.

Third Possible Error:

Conceptual confusion: Inverting the scale factor

Some students might compute \(\mathrm{k} = \frac{\mathrm{AD}}{\mathrm{EH}} = \frac{2}{5}\) instead of \(\mathrm{k} = \frac{\mathrm{EH}}{\mathrm{AD}} = \frac{5}{2}\), thinking about the ratio backwards. This would lead to:

• \(\text{Area of EFGH} = 12 \times \left(\frac{2}{5}\right)^2 = 12 \times \frac{4}{25} = \frac{48}{25} \approx 1.92\)

This is much smaller than 12, which should trigger suspicion (the larger rectangle should have a larger area), but it doesn't match any answer choice, leading to confusion and guessing.

The Bottom Line:

The critical insight is understanding that area scales as the square of linear dimensions. This is a fundamental property of two-dimensional scaling that students often confuse with simple proportional relationships. The \(\mathrm{k}^2\) rule is what distinguishes this problem from basic proportion problems.