Question

1. kristen divides each convex polygon below into triangles by drawing all diagonals from a single vertex. she conjectures that the number of triangles is always two less than the number of sides of the convex polygon. a. is kristens conjecture reasonable? explain. b. provide an additional example that supports your answer to item 7a.

Explanation:

Step1: Analyze the concept

For a convex polygon with \(n\) sides, when we draw all diagonals from a single vertex, the number of non - adjacent vertices to a given vertex is \(n - 3\) (we exclude the two adjacent vertices and the vertex itself). The number of diagonals drawn from a single vertex is \(n - 3\), and the number of triangles formed is \(n-2\).

Step2: Consider the general case

Let \(n\) be the number of sides of a convex polygon. When we draw diagonals from a single vertex, the number of regions (triangles) formed is related to the number of line - segments (diagonals and sides adjacent to the vertex) emanating from that vertex. The number of triangles \(t=n - 2\).

Step3: Provide an example for part b

For a heptagon (\(n = 7\)), when we draw diagonals from a single vertex, the number of non - adjacent vertices is \(7-3=4\). The number of triangles formed by drawing diagonals from a single vertex is \(7 - 2=5\).

Answer:

a. Yes, Kristen's conjecture is reasonable. When we draw diagonals from a single vertex of a convex polygon with \(n\) sides, the number of triangles formed is \(n - 2\). This is because the number of diagonals from a single vertex is \(n-3\), and these diagonals along with the two sides adjacent to the vertex divide the polygon into \(n - 2\) triangles.
b. For a heptagon (7 - sided polygon), when diagonals are drawn from a single vertex, the number of triangles formed is \(7-2 = 5\).