In the xy-plane, point C has coordinates \((h, k)\). Point A has coordinates \((h + 1, k + \sqrt{102})\). Point...

1. TRANSLATE the problem information

• Given information:
  • Point C at \((h, k)\)
  • Point A at \((h + 1, k + \sqrt{102})\)
  • Point B lies on line \(y = k + \sqrt{102}\)
  • \(\angle ACB\) is a right angle

• What this tells us: A right angle means the vectors CA and CB are perpendicular, so their dot product equals zero.

2. INFER the approach using vectors

• Since we have coordinates and a perpendicular condition, we should:
  • Express the angle condition using vectors
  • Use the dot product property of perpendicular vectors
  • Apply the constraint that B lies on the given line

3. Set up the vectors

Vector \(\mathrm{CA} = \mathrm{A} - \mathrm{C}\)

\(= (h + 1, k + \sqrt{102}) - (h, k)\)

\(= (1, \sqrt{102})\)

Since B lies on \(y = k + \sqrt{102}\), we can write \(\mathrm{B} = (x, k + \sqrt{102})\) for some unknown x.

Vector \(\mathrm{CB} = \mathrm{B} - \mathrm{C}\)

\(= (x, k + \sqrt{102}) - (h, k)\)

\(= (x - h, \sqrt{102})\)

4. APPLY CONSTRAINTS using the dot product condition

For perpendicular vectors: \(\mathrm{CA} \cdot \mathrm{CB} = 0\)

\((1, \sqrt{102}) \cdot (x - h, \sqrt{102}) = 0\)

5. SIMPLIFY the dot product equation

\(1(x - h) + \sqrt{102}(\sqrt{102}) = 0\)

\(x - h + 102 = 0\)

\(x - h = -102\)

\(x = h - 102\)

So B is at \((h - 102, k + \sqrt{102})\)

6. Calculate the distance AB

A is at \((h + 1, k + \sqrt{102})\) and B is at \((h - 102, k + \sqrt{102})\)

Since both points have the same y-coordinate, the distance is just the difference in x-coordinates:

\(\mathrm{AB} = |(h - 102) - (h + 1)|\)

\(= |-102 - 1|\)

\(= |-103|\)

\(= 103\)

Answer: D. 103

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not recognize that "\(\angle ACB\) is a right angle" means the vectors CA and CB are perpendicular with dot product zero. Instead, they might try to use right triangle relationships or attempt to work directly with angles, which becomes much more complex with the given coordinates.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the dot product equation but make arithmetic errors when solving \(x - h + 102 = 0\), perhaps getting \(x - h = 102\) instead of \(x - h = -102\). This would place B at \((h + 102, k + \sqrt{102})\) instead of \((h - 102, k + \sqrt{102})\), leading to \(\mathrm{AB} = |102 - 1| = 101\), which isn't among the choices.

This causes them to get stuck and guess.

The Bottom Line:

This problem requires recognizing that geometric angle conditions translate to algebraic vector relationships. The key insight is that perpendicular vectors have zero dot product - without this connection, the coordinate approach becomes unnecessarily complicated.