1. **Problem Statement:** We need to find the total number of triangles of all sizes in a large triangle subdivided by vertical and horizontal lines. 2. **Understanding the Diagram:** The large triangle is divided vertically into 3 smaller triangles by 2 vertical lines. Additionally, there are 3 horizontal lines inside the large triangle intersecting these vertical lines. 3. **Key Insight:** Each horizontal line creates new smaller triangles by intersecting the vertical divisions. The total number of triangles includes all small, medium, and large triangles formed by these intersections. 4. **Formula and Approach:** For a triangle subdivided by $n$ vertical lines and $m$ horizontal lines inside it, the total number of triangles can be found by counting all possible triangles formed by choosing vertices from the grid points formed by these lines. 5. **Counting Triangles:** - The vertical lines divide the base into 3 segments, so there are 4 vertical points (including the vertices). - The 3 horizontal lines create 4 horizontal levels (including the base and the top vertex). 6. **Number of triangles formed:** The total number of triangles in such a grid is given by the formula for triangular numbers summing over the levels: $$\text{Total triangles} = \sum_{k=1}^{m+1} k^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$$ 7. **Explanation:** Each horizontal level $k$ contributes $k^2$ triangles because at that level, the number of smaller triangles formed is the square of the number of segments. 8. **Final Answer:** There are **30** triangles of all sizes in the diagram.