Point A is a vertex of an 8-sided polygon. The polygon has 8 sides of equal length and 8 angles of equal measure. When all possible diagonals are drawn from vertex A in the polygon, how many triangles are formed?

To solve this problem, let's analyze the given information step-by-step.

### Step 1: Understand the Problem
- We have a regular 8-sided polygon (an octagon) with 8 sides of equal length and 8 angles of equal measure.
- We need to find the number of triangles formed by drawing all possible diagonals from one vertex (Point A).

### Step 2: Key Observations
1. **Vertices of the Octagon:** An octagon has 8 vertices, labeled \( A_1, A_2, A_3, \ldots, A_8 \). We can consider vertex \( A_1 \) as point A.
   
2. **Diagonals from Vertex A:** Diagonals are lines connecting non-adjacent vertices. From \( A_1 \), diagonals can be drawn to vertices \( A_3, A_4, A_5, A_6, \) and \( A_7 \). The lines to \( A_2 \) and \( A_8 \) are sides of the octagon, not diagonals.

3. **Forming Triangles:** Each triangle formed must include vertex \( A_1 \). The remaining two vertices of the triangle must come from the other vertices of the octagon. 

### Step 3: Counting the Number of Triangles
- The total number of vertices available other than \( A_1 \) is 7 (\( A_2, A_3, \ldots, A_8 \)).
- To form a triangle, we need to select 2 vertices from these 7. The number of ways to choose 2 vertices from 7 is given by the combination formula:

\[
\binom{7}{2} = \frac{7 \times 6}{2} = 21
\]

### Step 4: Conclusion
Therefore, the number of triangles formed when all possible diagonals are drawn from vertex \( A_1 \) in the polygon is **21**.

### Summary
By analyzing the properties of a regular octagon and using combinatorial methods, we determined that 21 triangles can be formed by drawing all possible diagonals from a single vertex.

Would you like more details on any of these steps or have any other questions?

### Related Questions:
1. How many diagonals are there in a regular 8-sided polygon in total?
2. How would the number of triangles change if the polygon had 10 sides?
3. What is the formula for calculating the number of diagonals in any regular polygon?
4. How many quadrilaterals can be formed by choosing any four vertices of a regular octagon?
5. How does the internal angle measure of a regular polygon change with the number of sides?

### Tip:
Remember that when working with regular polygons, symmetry and patterns often help simplify complex problems!

Explore how to calculate the number of triangles formed when all possible diagonals are drawn from a vertex in a regular octagon. This problem involves geometry and combinatorial mathematics, utilizing the combination formula to determine the number of triangles. Suitable for students in Grades 8-10.

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