In a circle with radius 6 inches, arc AB has length 4pi inches. Arc BC has length 5pi inches and...

1. TRANSLATE the problem information

• Given information:
  • Circle radius: \(\mathrm{6}\) inches
  • Arc AB length: \(\mathrm{4\pi}\) inches
  • Arc BC length: \(\mathrm{5\pi}\) inches
  • Arcs AB and BC share endpoint B and extend in same direction
• Need to find: Central angle measure for arc AC in degrees

2. INFER the solution approach

• Since we have arc lengths and radius, we can use the arc length formula \(\mathrm{s = r\theta}\) to find central angles
• Because the arcs extend in the same direction from shared point B, arc AC represents the combined span of both arcs
• This means the central angle for AC equals the sum of central angles for AB and BC

3. SIMPLIFY to find central angle for arc AB

• Using \(\mathrm{s = r\theta}\): \(\mathrm{4\pi = 6\theta}\)
• Central angle for AB = \(\mathrm{\frac{4\pi}{6} = \frac{2\pi}{3}}\) radians

4. SIMPLIFY to find central angle for arc BC

• Using \(\mathrm{s = r\theta}\): \(\mathrm{5\pi = 6\theta}\)
• Central angle for BC = \(\mathrm{\frac{5\pi}{6}}\) radians

5. SIMPLIFY to find total central angle for arc AC

• Central angle for AC = \(\mathrm{\frac{2\pi}{3} + \frac{5\pi}{6}}\)
• Convert to common denominator: \(\mathrm{\frac{2\pi}{3} = \frac{4\pi}{6}}\)
• Central angle for AC = \(\mathrm{\frac{4\pi}{6} + \frac{5\pi}{6} = \frac{9\pi}{6} = \frac{3\pi}{2}}\) radians

6. TRANSLATE from radians to degrees

• \(\mathrm{\frac{3\pi}{2} \times \frac{180°}{\pi} = 3 \times 90° = 270°}\)

Answer: C (270)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "extending in the same direction" means the arcs are consecutive and their central angles should be added together.

Instead, they might try to subtract the angles or use some other relationship, leading to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when adding the fractions \(\mathrm{\frac{2\pi}{3} + \frac{5\pi}{6}}\), often getting \(\mathrm{\frac{7\pi}{9}}\) instead of \(\mathrm{\frac{3\pi}{2}}\).

Converting \(\mathrm{\frac{7\pi}{9}}\) to degrees gives approximately \(\mathrm{140°}\), which doesn't match any answer choice. This leads to confusion and guessing, or they might select Choice A (135) as the closest value.

The Bottom Line:

This problem requires both geometric visualization (understanding what "same direction" means for consecutive arcs) and careful fraction arithmetic. Students who miss either component will struggle to reach the correct answer systematically.