Problem 6 · Double counting in a triangulation · 4 points
Made by AI and not reviewed by a real person.
A convex polygon is cut into triangles by noncrossing diagonals. Four triangles have no side on the polygon’s boundary, and five have exactly one boundary side. How many sides does the polygon have?
- 15
- 17
- 16
- 18
- 14
Show a hint
- Think about what the triangles tell you about the polygon’s original boundary, not just the total number of triangles.
- Let x be the number of triangles with two boundary sides. Express both the total number of triangles and the number of polygon sides in terms of x.
Show the answer and solution
Answer: B — 17
Let x triangles have two boundary sides. There are 4 + 5 + x = 9 + x triangles, so the polygon has 11 + x sides. Counting its boundary sides through the triangles instead gives 5 + 2x. Thus 11 + x = 5 + 2x, so x = 6 and the polygon has 17 sides. The tempting answer 15 is the number of triangles, not sides.