The areas of two similar polygons, Polygon A and Polygon B, are 16 square centimeters and 36 square centimeters, respectively....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Area\:of\:Polygon\:A = 16\:cm^2}\)
  • \(\mathrm{Area\:of\:Polygon\:B = 36\:cm^2}\)
  • \(\mathrm{One\:side\:length\:of\:Polygon\:A = 4\:cm}\)
  • Need: corresponding side length of Polygon B

2. INFER the mathematical relationship needed

• Since these are similar polygons, we can use the area scaling rule
• Key insight: The ratio of areas equals the square of the ratio of corresponding sides
• Strategy: Find the area ratio first, then work backwards to get the side length ratio

3. SIMPLIFY to find the area ratio

• \(\mathrm{Area\:ratio = \frac{Area\:of\:B}{Area\:of\:A} = \frac{36}{16} = \frac{9}{4}}\)

4. TRANSLATE the scaling relationship into an equation

• If \(\mathrm{s_A}\) and \(\mathrm{s_B}\) are corresponding side lengths: \(\mathrm{\left(\frac{s_B}{s_A}\right)^2 = \frac{9}{4}}\)

5. SIMPLIFY to find the side length ratio

• Take square root of both sides: \(\mathrm{\frac{s_B}{s_A} = \sqrt{\frac{9}{4}} = \frac{3}{2}}\)
• This means Polygon B's sides are \(\mathrm{\frac{3}{2}}\) times as long as Polygon A's sides

6. SIMPLIFY to find the actual side length

• We know \(\mathrm{s_A = 4}\), so: \(\mathrm{\frac{s_B}{4} = \frac{3}{2}}\)
• Therefore: \(\mathrm{s_B = 4 \times \frac{3}{2} = 6}\)

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to use the area ratio directly as the side length ratio, thinking that if areas have ratio 36:16, then sides have the same ratio.

Using \(\mathrm{\frac{36}{16} = \frac{9}{4} = 2.25}\), they calculate: \(\mathrm{side\:length = 4 \times 2.25 = 9}\)

This leads them to select Choice C (9).

Second Most Common Error:

Missing conceptual knowledge about area scaling: Students remember that similar figures have proportional sides but forget that areas scale by the square of the side ratio. They might set up the relationship backwards or get confused about which ratio to square.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

The key challenge is recognizing that area relationships and side length relationships are different for similar figures - you must take the square root of the area ratio to find the side length ratio. Students who miss this connection will use the area ratio directly and get the wrong answer.