Question

Let "n" represent the number of sides of any polygon and write an expression to represent the number of triangles that an n-gon can be divided into by using non-overlapping diagonals. (Hint: For each shape in the table, compare the number of sides in the polygon to the number of triangles, look for a pattern, and then write the expression.)

Answer 1

The expression to represent the number of triangles that can be formed by dividing an n-gon with non-overlapping diagonals is T = n - 2, where T is the number of triangles and n is the number of sides in the polygon.

Step-by-step explanation:

The question asks about writing an expression to represent the number of triangles a polygon can be divided into using non-overlapping diagonals.

For a polygon with n sides (n-gon), we know that a triangle (3-gon) cannot be divided into smaller triangles by its diagonals, which sets a base case. Beginning with a quadrilateral (4-gon), we can see that it can be divided into two triangles. For a pentagon (5-gon), it can be divided into three triangles, and a hexagon (6-gon) can be divided into four triangles, and so on. It appears there is a pattern where the number of triangles is always n-2 for any polygon with n sides.

Therefore, the expression to represent the number of triangles that can be formed by dividing an n-gon with non-overlapping diagonals is T = n - 2, where T is the number of triangles and n is the number of sides in the polygon.