To solve this problem and count the number of triangles in the given triangular figure, I will break it down step by step. The image shows a larger triangle subdivided into smaller triangles arranged in rows.

Breakdown:

1. Count individual small triangles: Each row of the large triangle contains multiple small triangles. For simplicity, we start by counting the smallest individual triangles.

   • Top row: 1 small triangle.
   • Second row: 2 triangles (since each larger triangle is divided into smaller ones).
   • Third row: 3 triangles.
   • Fourth row: 4 triangles. Total small triangles: $1 + 2 + 3 + 4 = 10$ small triangles.

2. Count larger triangles: In addition to the smallest triangles, there are larger triangles formed by combining smaller ones.

   • Triangles formed by combining 2 adjacent small triangles in each row:

     • Second row: 1 larger triangle.
     • Third row: 2 larger triangles.
     • Fourth row: 3 larger triangles. Total triangles of this type: $1 + 2 + 3 = 6$ larger triangles.

   • Triangles formed by combining 4 small triangles (a square of 4 triangles):

     • Third row: 1 larger triangle.
     • Fourth row: 3 larger triangles. Total triangles of this type: $1 + 3 = 4$ larger triangles.

3. Count the entire large triangle: The whole triangle itself is also counted as a triangle, adding 1 more to the total.

Final Calculation:

• Smallest triangles: 10
• Medium-sized triangles: 6 (formed by combining 2 small triangles)
• Larger triangles: 4 (formed by combining 4 small triangles)
• Entire large triangle: 1

Thus, the total number of triangles in the figure is:

$10 + 6 + 4 + 1 = 21$

So, the number of triangles in the given figure is 21.

Would you like further explanation or a breakdown of any specific step? Here are some related questions:

1. How do different shapes form composite triangles?
2. What are some methods for counting geometric patterns efficiently?
3. How can you generalize the process for counting triangles in other figures?
4. Could symmetry help simplify problems like this one?
5. Are there any formulas that directly calculate the number of triangles in figures like this?

Tip:

When counting triangles in complex figures, work systematically by identifying and classifying triangles of different sizes to avoid missing any.