https://file.aiquickdraw.com/mathbot/file/1731631099897s07m5ftr.jpg

Perhatikan gambar berikut.

Jika suatu segitiga dibentuk dengan menggunakan 3 titik (titik titik berbentuk segiempat). Berapa banyak segitiga yang dapat dibentuk

To determine the number of triangles that can be formed by selecting three points from the given grid of dots, let's analyze the setup.

### Step-by-Step Solution

1. **Identify Points and Configuration**:
   - The diagram shows a grid of dots in a 4x4 arrangement, resulting in 16 points.

2. **Selecting Combinations**:
   - To form a triangle, we need to select three points from these 16. The number of ways to choose 3 points out of 16 is given by the combination formula:
     \[
     \binom{16}{3} = \frac{16!}{3!(16 - 3)!} = \frac{16 \times 15 \times 14}{6} = 560
     \]
   - This gives us the total number of possible combinations of three points.

3. **Filtering Collinear Combinations**:
   - Not all combinations will result in a triangle. If three points are collinear, they cannot form a triangle.
   - In a 4x4 grid, collinear sets include horizontal lines, vertical lines, and some diagonal lines.

4. **Count Collinear Sets**:
   - **Horizontal and Vertical Lines**: Each row and each column has 4 points. We can select 3 points out of these 4 in each row or column.
     - There are 4 rows and 4 columns, each providing \(\binom{4}{3} = 4\) collinear combinations.
     - Total collinear combinations for rows and columns: \(4 \times 4 + 4 \times 4 = 32\).
   - **Diagonals**: We also need to consider the diagonals:
     - There are two main diagonals with 4 points each, giving \(2 \times \binom{4}{3} = 8\) collinear combinations.
     - Shorter diagonals contain fewer than 3 points and do not affect our count.

5. **Calculate Non-Collinear Combinations**:
   - Total combinations of three points: 560
   - Total collinear combinations: 32 (rows/columns) + 8 (diagonals) = 40
   - Therefore, the number of triangles that can be formed is:
     \[
     560 - 40 = 520
     \]

### Answer
The number of triangles that can be formed by selecting three points from the grid is **520**.

Would you like further details or have any questions?

Here are 5 related questions that expand on this concept:
1. How would the answer change if the grid size were increased to a 5x5 arrangement?
2. What if we considered only equilateral triangles? How many could we form?
3. How do we approach the problem if the points are arranged in a circular pattern instead of a grid?
4. How can we extend this method to find quadrilaterals instead of triangles?
5. What are the mathematical principles behind identifying collinear points in any arrangement?

**Tip**: When analyzing point arrangements, look for patterns of symmetry and regularity to simplify counting distinct geometric shapes.

Perhatikan gambar berikut.

Jika suatu segitiga dibentuk dengan menggunakan 3 titik (titik-titik berbentuk segiempat). Berapa banyak segitiga yang dapat dibentuk?

Discover how to calculate the number of triangles that can be formed by selecting three non-collinear points from a 4x4 grid of dots. Learn key combinatorics and geometry concepts and understand how to identify collinear points in grid configurations.

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