Explanation:

The given figure is a large triangle that is divided by lines drawn from its top vertex to the base, and also by a horizontal line parallel to the base. \nTo count the number of triangles systematically in such a figure, we can use a standard method: 1. **Count the number of segments on the base:** Identify the number of smaller segments formed on the main base (or any horizontal line) by the lines originating from the opposite vertex. In this figure, the lines from the top vertex divide the base into 3 segments. Let $n$ be the number of segments on the base. Here, $n=3$. 2. **Count the number of horizontal levels:** Identify the number of horizontal lines (including the main base) that act as bases for triangles. In this figure, there are two such horizontal lines: the main base and the internal horizontal line. Let $k$ be the number of horizontal levels. Here, $k=2$. 3. **Apply the formula:** The total number of triangles in such a configuration is given by the formula: $k \times \frac{n(n+1)}{2}$. * For a single horizontal base with $n$ segments, the number of triangles is $\frac{n(n+1)}{2}$. In our case, for $n=3$, this is $\frac{3(3+1)}{2} = \frac{3 \times 4}{2} = \frac{12}{2} = 6$. * Since there are $k=2$ horizontal levels, we multiply this by 2. Total triangles $= 2 \times 6 = 12$. **Alternatively, we can list them systematically:** **A. Triangles with their base on the bottommost horizontal line (main base):** * **Smallest triangles (1 segment base):** There are 3 such triangles. * **Medium triangles (2 segments base):** There are 2 such triangles (formed by combining two adjacent small triangles). * **Large triangles (3 segments base):** There is 1 such triangle (the largest one with the bottom base). * Total from the bottom base: $3 + 2 + 1 = 6$ triangles. **B. Triangles with their base on the middle horizontal line:** * Similarly, for the middle horizontal line, it is also divided into 3 segments by the lines originating from the top vertex. * **Smallest triangles (1 segment base):** There are 3 such triangles. * **Medium triangles (2 segments base):** There are 2 such triangles. * **Large triangles (3 segments base):** There is 1 such triangle (the largest one with the middle base). * Total from the middle horizontal line: $3 + 2 + 1 = 6$ triangles. **Total number of triangles = (Triangles from bottom base) + (Triangles from middle horizontal line)**\nTotal triangles $= 6 + 6 = 12$. \nLet's analyze why other options are incorrect: * **A. 9:** This undercounts the total number of triangles, likely missing combinations or triangles from one of the horizontal levels. * **B. 10:** This also undercounts, possibly by miscounting combinations or missing a level of triangles. * **D. 13:** This overcounts, as our systematic method and formula consistently yield 12 triangles without any ambiguity. \nThe correct count is 12.