Points A, B, and C are noncollinear in the plane. The distance from A to B is 8 units, and...

1. TRANSLATE the problem information

• Given information:
  • Points A, B, C are noncollinear
  • \(\mathrm{AB = 8}\) units
  • \(\mathrm{BC = 15}\) units
  • \(\mathrm{x = AC}\) (what we need to find bounds for)

2. INFER what noncollinear points mean

• Since A, B, and C are noncollinear, they form a triangle
• This means triangle inequality theorem applies
• For any triangle, each side length must be less than the sum of the other two sides AND greater than their absolute difference

3. APPLY CONSTRAINTS using triangle inequality

• Set up the three triangle inequality conditions:
  • \(\mathrm{AB + BC \gt AC}\) → \(\mathrm{8 + 15 \gt x}\) → \(\mathrm{x \lt 23}\)
  • \(\mathrm{AB + AC \gt BC}\) → \(\mathrm{8 + x \gt 15}\) → \(\mathrm{x \gt 7}\)
  • \(\mathrm{BC + AC \gt AB}\) → \(\mathrm{15 + x \gt 8}\) → \(\mathrm{x \gt -7}\) (always true for positive distances)

4. SIMPLIFY to find the final range

• From \(\mathrm{x \lt 23}\) and \(\mathrm{x \gt 7}\): \(\mathrm{7 \lt x \lt 23}\)
• This matches answer choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that "noncollinear points" means the points form a triangle, so triangle inequality applies. Instead, they might think this is just about distance relationships without geometric constraints.

Without this key insight, students often focus only on the obvious constraint that AC cannot exceed AB + BC = 23, leading them to select Choice A (\(\mathrm{x \lt 23}\)).

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students recognize triangle inequality applies but incorrectly use AB = 8 as the lower bound instead of \(\mathrm{|AB - BC| = |8 - 15| = 7}\). They reason that AC must be greater than AB since BC is the largest side.

This leads them to select Choice B (\(\mathrm{8 \lt x \lt 23}\)).

The Bottom Line:

This problem tests whether students can connect the geometric meaning of "noncollinear" to triangle formation and then systematically apply all triangle inequality constraints. The key insight is that the lower bound comes from the absolute difference, not just one of the given side lengths.