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

1. TRANSLATE the problem information

• Given information:
  • Circle with center C at coordinates \(\mathrm{(h, k)}\)
  • Point A at coordinates \(\mathrm{(h + 1, k +} \sqrt{102}\mathrm{)}\) lies on the circle
  • Point B also lies on the circle
  • Angle ACB is a right angle
  • Need to find: length of AB

2. INFER key geometric relationships

• Since both A and B lie on the circle with center C, this means:
  • CA is a radius of the circle
  • CB is also a radius of the circle
  • Therefore: \(\mathrm{CA = CB}\) (all radii of a circle are equal)

3. SIMPLIFY to find the radius length

• Use the distance formula to find CA:
  • \(\mathrm{CA} = \sqrt{[(\mathrm{h} + 1 - \mathrm{h})^2 + (\mathrm{k} + \sqrt{102} - \mathrm{k})^2]}\)
  • \(\mathrm{CA} = \sqrt{[1^2 + (\sqrt{102})^2]}\)
  • \(\mathrm{CA} = \sqrt{[1 + 102]} = \sqrt{103}\)
• Therefore: \(\mathrm{CA = CB =} \sqrt{103}\)

4. INFER the triangle type and setup

• Since \(\angle\mathrm{ACB}\) is a right angle, triangle ACB is a right triangle
• The two legs are CA and CB (both length \(\sqrt{103}\))
• The hypotenuse is AB (what we're looking for)

5. SIMPLIFY using the Pythagorean theorem

• Apply: \(\mathrm{AB}^2 = \mathrm{CA}^2 + \mathrm{CB}^2\)
• \(\mathrm{AB}^2 = (\sqrt{103})^2 + (\sqrt{103})^2\)
• \(\mathrm{AB}^2 = 103 + 103 = 206\)
• \(\mathrm{AB} = \sqrt{206}\)

Answer: A. \(\sqrt{206}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that CA and CB are both radii of the circle, so they're equal in length.

Without this insight, students might try to find coordinates for point B or use different approaches that lead nowhere. This leads to confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly find that the radius is \(\sqrt{103}\) and set up the Pythagorean theorem, but make calculation errors.

For example, they might calculate \(\mathrm{AB}^2 = (\sqrt{103})^2 + (\sqrt{103})^2 = \sqrt{103} + \sqrt{103} = 2\sqrt{103}\), forgetting to square the terms first. This leads them to think \(\mathrm{AB} = \sqrt{(2\sqrt{103})}\), which doesn't match any answer choice and causes them to guess.

The Bottom Line:

This problem tests your ability to connect coordinate geometry with circle properties and right triangles. The key insight is recognizing that points on a circle create equal radii, which gives you the equal legs needed for the Pythagorean theorem.