Choose the correct decomposition of the regular polygon into n-2 triangles

Question

GPT-4o mini · Answer

To decompose a regular polygon into $ n-2 $ triangles, we can use a method called triangulation. This method involves drawing diagonals from one vertex of the polygon to all other non-adjacent vertices. For a regular polygon with $ n $ sides, the steps to decompose it into $ n-2 $ triangles are as follows: 1. **Choose one vertex** of the polygon. Let's call this vertex $ A_1 $. 2. **Draw diagonals** from vertex $ A_1 $ to all other vertices except the two adjacent ones (let's call them $ A_2 $ and $ A_n $). This means you will draw diagonals to the vertices $ A_3, A_4, \ldots, A_{n-1} $. 3. Each of these diagonals will form a triangle with the sides of the polygon. By following this method, you will end up creating $ n-2 $ triangles: - Triangle $ A_1 A_2 A_3 $ - Triangle $ A_1 A_3 A_4 $ - Triangle $ A_1 A_4 A_5 $ - ... - Triangle $ A_1 A_{n-2} A_{n-1} $ In summary, the correct decomposition of a regular polygon into $ n-2 $ triangles can be accomplished by selecting one vertex and connecting it with diagonals to the non-adjacent vertices. This effectively divides the polygon into the desired number of triangles while ensuring that all angles are preserved, as each triangle is formed strictly by the sides of the polygon and the chosen diagonals.