Understanding the Problem: Triangles in a Regular Polygon

The question asks us to find the number of triangles formed by selecting vertices of a regular polygon with 10 sides (a decagon). The key constraint is that none of the sides of the formed triangle should be a side of the original polygon.

A regular polygon with 10 sides has 10 vertices.

Calculating Total Possible Triangles from Vertices

To form any triangle, we need to choose 3 distinct vertices from the 10 available vertices. The total number of ways to choose 3 vertices out of 10 is given by the combination formula:

Total triangles = $\binom{n}{k} = \binom{10}{3}$

Where n is the total number of vertices (10) and k is the number of vertices needed for a triangle (3).

Let's calculate the total number of triangles:

$\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120$

So, there are a total of 120 possible triangles that can be formed by joining the vertices of the 10-sided polygon.

Identifying Triangles Sharing Polygon Sides

The problem requires triangles that have no common side with the polygon. This means we need to subtract the triangles that *do* share sides with the polygon from the total number of triangles. Triangles formed by the vertices of the polygon can share either one side or two sides with the polygon.

Calculating Triangles Sharing Exactly One Side with the Polygon

A triangle shares exactly one side with the polygon if one of its sides is a side of the polygon, and the other two sides are diagonals of the polygon.

To count these triangles, we can follow these steps:

• First, choose one side of the 10-sided polygon. There are 10 sides. Let's say we choose the side connecting vertex V1 and V2.
• The third vertex of the triangle must be one of the remaining $10 - 2 = 8$ vertices.
• However, the third vertex cannot be adjacent to either V1 or V2, because that would create a triangle sharing two sides with the polygon (e.g., if we chose V3, triangle V1V2V3 shares V1V2 and V2V3). In a 10-gon, V3 is adjacent to V2 and V10 is adjacent to V1.
• So, we must exclude the vertices V1, V2, and their adjacent vertices (V3 and V10) that form sides of the polygon.
• The number of available vertices for the third point is $10 - (\text{V1, V2, V3, V10}) = 10 - 4 = 6$.
• For each of the 10 sides, there are 6 choices for the third vertex to form a triangle sharing exactly one side.

Number of triangles sharing exactly one side = (Number of sides) $\times$ (Number of choices for the third vertex)

Number of triangles sharing exactly one side = $10 \times 6 = 60$.

Calculating Triangles Sharing Exactly Two Sides with the Polygon

A triangle shares exactly two sides with the polygon if its vertices are three consecutive vertices of the polygon. For example, vertices V1, V2, and V3 form a triangle V1V2V3 which shares sides V1V2 and V2V3 with the polygon.

To count these triangles, we just need to count the number of sets of three consecutive vertices in the 10-sided polygon. These are (V1, V2, V3), (V2, V3, V4), ..., (V10, V1, V2).

There are exactly as many such triangles as there are vertices (or sides) in the polygon.

Number of triangles sharing exactly two sides = 10.

Finding Triangles with No Common Side

The number of triangles that have no common side with the polygon is obtained by subtracting the triangles that share one or two sides from the total number of triangles.

Number of no-common-side triangles = Total triangles - (Triangles sharing exactly one side) - (Triangles sharing exactly two sides)

Number of no-common-side triangles = $120 - 60 - 10 = 50$.

Summary of Triangle Counts in 10-gon

Final Answer: Number of Triangles with No Common Side

The number of triangles that can be formed by joining the vertices of the regular 10-sided polygon which have no common side with any of the sides of the polygon is 50.

Revision Table: Polygon Triangle Formulas

Additional Information: Geometric Counting and Combinatorics

This problem demonstrates how combinatorics can be applied to count geometric objects. The method used here, subtracting unwanted cases from the total, is a common strategy in counting problems. The formula for triangles with no common side in an n-gon, $\binom{n}{3} - n(n-4) - n$, is a useful result derived from this approach. This type of problem helps build a strong understanding of combinations and how constraints affect counting possibilities.