In the xy-plane, a circle has center C with coordinates \((\mathrm{h}, \mathrm{k})\). Point A lies on the circle and has...

1. TRANSLATE the problem information

• Given information:
  • Circle with center C at \((\mathrm{h}, \mathrm{k})\)
  • Point A on circle at \((\mathrm{h} - 2, \mathrm{k} + \sqrt{93})\)
  • \(\angle \mathrm{ADB} = 60°\) where A, B, D are points on the circle
  • Need to find length AB

2. INFER what we need first

• To find a chord length, we need the radius and the central angle
• We can get the radius from the center and point A coordinates
• The 60° angle is key, but we need to determine what type of angle it is

3. SIMPLIFY to find the radius

• Since A is on the circle, radius = distance from center C to point A
• \(\mathrm{r} = \sqrt{[(\mathrm{h} - 2 - \mathrm{h})^2 + (\mathrm{k} + \sqrt{93} - \mathrm{k})^2]}\)
• \(= \sqrt{[(-2)^2 + (\sqrt{93})^2]}\)
• \(= \sqrt{[4 + 93]}\)
• \(= \sqrt{97}\)

4. INFER the relationship between angles

• \(\angle\mathrm{ADB}\) has its vertex D on the circle with sides passing through A and B
• This makes it an inscribed angle subtending arc AB
• By inscribed angle theorem: central angle \(\angle\mathrm{ACB} = 2 \times 60° = 120°\)

5. SIMPLIFY using the chord length formula

• For chord AB with central angle 120°:
• \(\mathrm{AB} = 2\mathrm{r} \sin(120°/2)\)
• \(= 2\mathrm{r} \sin(60°)\)
• Since \(\sin(60°) = \sqrt{3}/2\):
• \(\mathrm{AB} = 2\mathrm{r} \times (\sqrt{3}/2)\)
• \(= \mathrm{r}\sqrt{3}\)
• \(\mathrm{AB} = \sqrt{97} \times \sqrt{3}\)
• \(= \sqrt{291}\)

Answer: \((\mathrm{D}) \sqrt{291}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(\angle\mathrm{ADB}\) is an inscribed angle, and instead treating the 60° as the central angle directly.

Students might think: "If the angle is 60°, then I can use the chord formula with \(\theta = 60°\)." This gives \(\mathrm{AB} = 2\mathrm{r} \sin(30°)\)

\(= 2\sqrt{97} \times (1/2)\)

\(= \sqrt{97}\), leading them to select Choice \((\mathrm{B}) \sqrt{97}\).

Second Most Common Error:

Poor SIMPLIFY execution: Making calculation errors when finding the radius or in the final multiplication step.

Some students might correctly set up \(\mathrm{r}\sqrt{3}\) but then compute \(\sqrt{97} \times \sqrt{3}\) incorrectly, perhaps getting \(\sqrt{97} + \sqrt{3}\) instead of \(\sqrt{(97 \times 3)}\), or make errors in the distance formula calculation. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires recognizing the inscribed angle theorem - that's the key insight that separates it from a straightforward chord calculation. Without this recognition, students will use the wrong central angle and get a plausible but incorrect answer.