A regular decagon is inscribed in a circle with center O. Vertices A and B are positioned so that, counting...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A regular decagon is inscribed in a circle with center \(\mathrm{O}\). Vertices \(\mathrm{A}\) and \(\mathrm{B}\) are positioned so that, counting along the shorter arc from \(\mathrm{A}\) to \(\mathrm{B}\), there are exactly two intermediate vertices. In triangle \(\mathrm{AOB}\), what is the measure of \(\angle\mathrm{OBA}\), in degrees? Enter your answer as an integer; do not include the degree symbol.
1. TRANSLATE the problem setup
- Given information:
- Regular decagon inscribed in circle with center O
- Vertices A and B with exactly 2 intermediate vertices between them (on shorter arc)
- Need to find \(\angle\mathrm{OBA}\) in triangle AOB
2. INFER the central angle per vertex
- Since the decagon is regular, all 10 vertices are equally spaced around the circle
- Each consecutive pair of vertices subtends the same central angle
- Central angle per vertex = \(360°/10 = 36°\)
3. INFER the relationship between A and B
- "Exactly 2 intermediate vertices" means we count: A → vertex₁ → vertex₂ → B
- This represents 3 "steps" around the decagon
- Therefore, central angle \(\angle\mathrm{AOB} = 3 \times 36° = 108°\)
4. INFER the triangle type
- Both OA and OB are radii of the same circle
- Since all radii are equal: \(\mathrm{OA} = \mathrm{OB}\)
- Triangle AOB is isosceles with vertex angle \(\angle\mathrm{AOB} = 108°\)
5. SIMPLIFY to find the base angle
- In isosceles triangle AOB, the base angles are equal: \(\angle\mathrm{OAB} = \angle\mathrm{OBA}\)
- Sum of triangle angles: \(\angle\mathrm{AOB} + \angle\mathrm{OAB} + \angle\mathrm{OBA} = 180°\)
- Substituting: \(108° + \angle\mathrm{OBA} + \angle\mathrm{OBA} = 180°\)
- Solving: \(108° + 2\angle\mathrm{OBA} = 180°\)
- Therefore: \(2\angle\mathrm{OBA} = 72°\), so \(\angle\mathrm{OBA} = 36°\)
Answer: 36
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students may miscount the separation between vertices A and B, thinking "2 intermediate vertices" means \(\angle\mathrm{AOB} = 2 \times 36° = 72°\) instead of recognizing it represents 3 central angle steps for \(\angle\mathrm{AOB} = 108°\).
When they use 72° as the vertex angle, they calculate: \(\angle\mathrm{OBA} = (180° - 72°)/2 = 54°\). This leads them to select an incorrect answer or get confused when 54 isn't an option.
Second Most Common Error:
Missing conceptual knowledge: Students may not recognize that OA and OB are equal radii, failing to identify triangle AOB as isosceles. Without this insight, they cannot use the equal base angles property and may get stuck trying to find additional information to solve the triangle.
This causes them to get stuck and abandon systematic solution, leading to guessing.
The Bottom Line:
This problem tests spatial reasoning about regular polygons and requires connecting multiple geometric concepts: central angles, isosceles triangles, and triangle angle relationships. The key insight is recognizing that "2 intermediate vertices" creates a 3-step separation around the decagon.