23.4.1. Defining Triangulations

Try these first

1. 1.

   What is the definition of triangulation?

   Hint

   Think about how you could separate a polygon into triangles.

2. 2.

   How do you express the number of ways to triangulate a polygon with n + 2 sides?

   Hint

   Consider using the variable T followed by n.

3. 3.

   What is the primary purpose of triangulation in geometry?

   • To divide shapes into triangles
   • To measure areas
   • To create non-overlapping paths

   Hint

   Think about how shapes are utilized in mathematical processes.

4. 4.

   True or False: The number of triangulations of a polygon corresponds to the nth Catalan number.

   • True
   • False

   Hint

   Remember the bijection between triangulations and parentheses.

5. 5.

   Given a hexagon, determine the number of triangulations and explain your reasoning.

   Hint

   Calculate using previously established formulas.

6. 6.

   Prove that the recurrence relation for T_n leads to the nth Catalan number.

   Hint

   Use induction as a proof strategy.