A circle has center O, and points A and B lie on the circle. The measure of arc AB is...

1. TRANSLATE the problem information

• Given information:
  • Arc AB measures 45°
  • Arc AB has length 3 inches
  • Need to find the circumference of the circle

2. INFER the key relationship

• The crucial insight: Arc length is proportional to arc measure
• This means: larger arc measure → longer arc length
• A full circle measures 360°, so we can set up a proportion

3. INFER the proportion setup

• If 45° gives us 3 inches of arc length
• Then 360° (the full circle) gives us the circumference
• Proportion: \(\frac{45°}{360°} = \frac{3 \text{ inches}}{\text{Circumference}}\)

4. SIMPLIFY the proportion

• \(\frac{45°}{360°} = \frac{3}{\mathrm{C}}\)
• Reduce the fraction: \(\frac{45}{360} = \frac{1}{8}\)
• So: \(\frac{1}{8} = \frac{3}{\mathrm{C}}\)

5. SIMPLIFY to find the circumference

• Cross multiply: \(1 × \mathrm{C} = 8 × 3\)
• \(\mathrm{C} = 24 \text{ inches}\)

Answer: D. 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the proportional relationship between arc measure and arc length. They might think the 3-inch arc length IS the answer since it's the only length given in the problem.

This may lead them to select Choice A (3).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the proportion correctly but make calculation errors. For example, they might incorrectly solve \(\frac{1}{8} = \frac{3}{\mathrm{C}}\) by thinking \(\mathrm{C} = \frac{3}{8}\) instead of \(\mathrm{C} = 3 × 8\), or make other arithmetic mistakes.

This causes them to get confused and potentially guess among the remaining choices.

The Bottom Line:

The key insight is recognizing that this is fundamentally a proportion problem. The arc's measure (45°) compared to a full circle (360°) must equal the arc's length (3 inches) compared to the full circumference. Without this proportional thinking, students either guess or focus only on the given numbers without seeing the bigger relationship.