Figure A and figure B are both regular polygons. The sum of the perimeter of figure A and the perimeter...

1. TRANSLATE the problem information

• Given information:
  • Figure A and Figure B are both regular polygons
  • Sum of their perimeters is 63 inches
  • Equation \(3\mathrm{x} + 6\mathrm{y} = 63\) represents this situation
  • \(\mathrm{x}\) = number of sides of figure A
  • \(\mathrm{y}\) = number of sides of figure B

• What this tells us: The equation must represent the total perimeter as the sum of both individual perimeters.

2. INFER the meaning of each term in the equation

• Since \(3\mathrm{x} + 6\mathrm{y} = 63\) represents the sum of perimeters:
  • The term \(3\mathrm{x}\) must represent the perimeter of figure A
  • The term \(6\mathrm{y}\) must represent the perimeter of figure B

• For regular polygons: \(\mathrm{perimeter} = (\mathrm{number\ of\ sides}) \times (\mathrm{side\ length})\)

3. INFER what the coefficients represent

• For figure A: \(\mathrm{perimeter} = \mathrm{x} \times (\mathrm{side\ length\ of\ A}) = 3\mathrm{x}\)
  • This means each side of figure A has length 3 inches

• For figure B: \(\mathrm{perimeter} = \mathrm{y} \times (\mathrm{side\ length\ of\ B}) = 6\mathrm{y}\)
  • This means each side of figure B has length 6 inches

• Therefore, the coefficient 6 represents the side length of figure B.

Answer: A. Each side of figure B has a length of 6 inches.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse what the variables and coefficients represent in the equation context.

They might think the coefficient 6 represents the number of sides rather than recognizing it as part of the perimeter calculation (number of sides × side length). This confusion about the structure of the equation leads them to select Choice B (The number of sides of figure B is 6).

Second Most Common Error:

Inadequate INFER reasoning: Students may correctly identify that \(3\mathrm{x}\) represents figure A's perimeter but then incorrectly assume the coefficient 3 applies to both figures.

They fail to make the connection that different coefficients (3 and 6) indicate different side lengths for the two polygons. This may lead them to select Choice C (Each side of figure A has a length of 6 inches).

The Bottom Line:

This problem requires students to work backward from an equation to understand the physical meaning of its components. The key insight is recognizing that in the context of perimeter = sides × side length, the coefficients must represent the side lengths of each respective polygon.