From a point A outside a circle, two segments AT and AS are drawn that are tangent to the circle...

1. TRANSLATE the problem information

• Given information:
  • Point A is outside a circle
  • AT and AS are tangent to the circle at points T and S
  • \(\mathrm{AT = 9}\) units
  • Perimeter of triangle ATS = 31 units
  • Find: length of TS

2. INFER the key geometric relationship

• Since AT and AS are both tangent segments drawn from the same external point A to the circle, they must be equal in length
• This gives us: \(\mathrm{AT = AS = 9}\) units

3. TRANSLATE the perimeter condition into an equation

• Perimeter means the sum of all three sides of triangle ATS
• So: \(\mathrm{AT + AS + TS = 31}\)

4. SIMPLIFY to solve for TS

• Substitute the known values: \(\mathrm{9 + 9 + TS = 31}\)
• Combine: \(\mathrm{18 + TS = 31}\)
• Solve: \(\mathrm{TS = 31 - 18 = 13}\)

Answer: C (13)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing that tangent segments from an external point are equal

Students might think AT and AS are different lengths and try to work with two unknowns. Without realizing that \(\mathrm{AT = AS = 9}\), they cannot set up a solvable equation. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak INFER skill: Recognizing the problem involves tangents but not connecting it to the equal length property

Some students might know about tangent properties in general but fail to apply the specific property that tangent segments from the same external point are equal. They might try to use other tangent properties (like perpendicularity to radii) that don't directly help solve this problem. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students know and can apply the fundamental property of tangent segments. Once that insight clicks, the algebra is straightforward - but without it, the problem is unsolvable.