A triangular road sign is reproduced at a larger size, creating a new sign that is similar to the original....

1. TRANSLATE the problem information

• Given information:
  • Original sign perimeter: 56 inches
  • Enlarged sign perimeter: 196 inches
  • One side of original sign: 14 inches
  • Signs are similar (same shape, different size)
• What we need: length of corresponding side on enlarged sign

2. INFER the key relationship

• Since the triangular signs are similar, all corresponding linear measurements scale by the same factor
• The scale factor can be found using any pair of corresponding linear measurements
• Perimeter is a linear measurement, so: \(\mathrm{scale\, factor} = \frac{\mathrm{enlarged\, perimeter}}{\mathrm{original\, perimeter}}\)

3. SIMPLIFY to find the scale factor

• \(\mathrm{Scale\, factor} = \frac{196}{56} = 3.5\) (use calculator)
• This means every linear dimension of the enlarged sign is 3.5 times the corresponding dimension of the original

4. SIMPLIFY to find the corresponding side

• \(\mathrm{Corresponding\, side} = \mathrm{original\, side} \times \mathrm{scale\, factor}\)
• \(\mathrm{Corresponding\, side} = 14 \times 3.5 = 49\, \mathrm{inches}\) (use calculator)

Answer: 49 inches (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that perimeter ratio equals the scale factor for individual sides. They might think they need to find the individual side lengths of the original triangle first, or they might not understand that similar figures scale uniformly.

This leads to confusion about how to connect the perimeter information to the side length question, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the scale factor from perimeter ratio, but make arithmetic errors. For example, calculating \(\frac{196}{56}\) incorrectly as 2.5 instead of 3.5, which would give \(14 \times 2.5 = 35\).

This may lead them to select Choice B (35).

The Bottom Line:

The key insight is recognizing that similarity means uniform scaling - the ratio of any corresponding linear measurements (including perimeters) gives you the scale factor for all linear measurements. Once you have that scale factor, applying it is straightforward multiplication.