6:49 C .1ll 56؛ 75 deltamath.com Test Overview 3 $$ \approx $$ DeltaMath Previous Question Next Question Question 3 Using diagonals from a common vertex, how many triangles could be formed from a decagon? Answer $$ : \because: $$ Attempt 2 out of 5 $$ \square $$ Submit Answer Copyright ©2025 DeltaMath.com All Rights Reserved. Terms \& Policies

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To determine how many triangles can be formed using diagonals from a common vertex in a decagon (a 10-sided polygon), let's break down the problem step by step.

### Step 1: Understanding the Decagon

A decagon has **10 vertices**. For any given vertex:

- It is connected to **2 adjacent vertices** via the sides of the decagon.
- It can be connected to the remaining **7 vertices** via diagonals.

### Step 2: Choosing the Common Vertex

Let's select one vertex as the **common vertex** (let's call it **A**) from which all diagonals will be drawn.

### Step 3: Drawing Diagonals from Vertex A

From vertex **A**, you can draw diagonals to **7 other vertices** (since connections to the adjacent vertices are sides, not diagonals).

### Step 4: Forming Triangles

To form a triangle using these diagonals:

1. **Select any two diagonals** from the 7 available. Each pair of diagonals will form a triangle with vertex **A**.
2. The number of ways to choose 2 diagonals from 7 is given by the combination formula:

$$
   \text{Number of triangles} = \binom{7}{2}
   $$

3. Calculating this:

$$
   \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21
   $$

### Conclusion

**21 triangles** can be formed using diagonals from a common vertex in a decagon.

## **Answer:** 21
21 triangles can be formed using diagonals from a common vertex in a decagon.

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