A total of 2 squares each have side length r. A total of 6 equilateral triangles each have side length...

1. TRANSLATE the problem information

• Given information:
  • 2 squares, each with side length \(\mathrm{r}\)
  • 6 equilateral triangles, each with side length \(\mathrm{t}\)
  • Total perimeter of all shapes = 210
• What this tells us: We need to find total perimeter from squares + total perimeter from triangles

2. INFER the approach

• Strategy: Calculate the perimeter contribution from each shape type separately, then add them
• We'll need to use perimeter formulas for squares and triangles

3. SIMPLIFY to find square perimeter contribution

• Perimeter of one square = \(4\mathrm{r}\)
• Total from 2 squares = \(2 \times 4\mathrm{r} = 8\mathrm{r}\)

4. SIMPLIFY to find triangle perimeter contribution

• Perimeter of one equilateral triangle = \(3\mathrm{t}\)
• Total from 6 triangles = \(6 \times 3\mathrm{t} = 18\mathrm{t}\)

5. INFER the final equation

• Total perimeter = \(8\mathrm{r} + 18\mathrm{t}\)
• Since total perimeter = 210: \(8\mathrm{r} + 18\mathrm{t} = 210\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret what quantities need to be calculated, focusing on individual side lengths rather than total perimeters.

They might think the equation should just use the side lengths directly (\(\mathrm{r}\) and \(\mathrm{t}\)) without calculating perimeters, leading them to create equations like \(2\mathrm{r} + 6\mathrm{t} = 210\) based on the number of shapes times their side lengths.

This may lead them to select Choice B (\(2\mathrm{r} + 6\mathrm{t} = 210\))

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for perimeters but make calculation errors when multiplying.

Common mistakes include calculating \(2 \times 4\mathrm{r} = 6\mathrm{r}\) instead of \(8\mathrm{r}\), or \(6 \times 3\mathrm{t} = 24\mathrm{t}\) instead of \(18\mathrm{t}\). These arithmetic errors lead to incorrect coefficients in the final equation.

This may lead them to select Choice A (\(6\mathrm{r} + 24\mathrm{t} = 210\)) or other incorrect options

The Bottom Line:

This problem requires careful translation of word quantities into mathematical expressions, plus accurate application of basic perimeter formulas. Students must distinguish between individual measurements and total contributions from multiple shapes.