QUESTION STEM:Hexagon H and hexagon J are regular and similar.The perimeter of H is 166 times the perimeter of J.If...

1. TRANSLATE the problem information

• Given information:
  • Hexagon H and hexagon J are regular and similar
  • Perimeter of H = 166 × Perimeter of J
  • Area of H = k × Area of J
  • Find the value of k

2. INFER the relationship between perimeter and linear scale factor

• Since the hexagons are similar, all corresponding linear dimensions are proportional
• Perimeter is a linear measurement (it's the sum of side lengths)
• Therefore: \(\mathrm{perimeter\ ratio = linear\ scale\ factor = 166}\)

3. INFER how area scales for similar figures

• For similar figures, area scales as the square of the linear scale factor
• If \(\mathrm{linear\ scale\ factor = 166}\), then \(\mathrm{area\ scale\ factor = 166^2}\)
• Therefore: \(\mathrm{k = 166^2}\)

4. SIMPLIFY to find the numerical value

• Calculate \(\mathrm{166^2}\)
• \(\mathrm{166^2 = (160 + 6)^2}\)
  \(\mathrm{= 160^2 + 2(160)(6) + 6^2}\)
  \(\mathrm{= 25600 + 1920 + 36 = 27556}\) (use calculator)

Answer: 27556

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that perimeter ratio equals linear scale factor

Many students see "perimeter of H is 166 times perimeter of J" and think this means the area is also 166 times larger. They confuse linear scaling with area scaling, leading them to answer \(\mathrm{k = 166}\). This leads to confusion since 166 isn't the correct answer, causing them to abandon systematic solution and start guessing.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember the area scaling rule for similar figures

Some students correctly identify that the linear scale factor is 166 but then don't know how area scales. They might try to find a pattern or guess that area scales linearly too, rather than as the square of the scale factor. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students understand the fundamental difference between how linear dimensions and areas scale in similar figures. The key insight is connecting perimeter (linear) to the scale factor, then applying the quadratic relationship for area scaling.