For triangle-counting problems, always use a system. Classify triangles by size, orientation, or type (equilateral, isosceles). Start with the smallest and work your way up. Mark triangles as you count them to avoid errors. Be aware that grids can often contain non-obvious, non-equilateral triangles.
Step 1: Understanding the Concept:
This is a classic visual puzzle that requires systematically identifying and counting all triangles of different sizes and orientations within a complex geometric figure. The figure is a rectangular grid composed of equilateral triangles.
Step 2: Detailed Explanation:
To avoid missing any triangles or double-counting, we will classify them by their size (based on side length) and orientation (pointing up or pointing down). Let the side length of the smallest triangle be 1 unit. 1. Triangles of Side Length 1:
Pointing Up: There are 6 such triangles with their bases on the bottom line of the figure.
Pointing Down: There are 6 such triangles with their bases on the top line of the figure.
Total Size-1 Triangles = \(6 + 6 = 12\). 2. Triangles of Side Length 2:
Pointing Up: These triangles have a base of length 2 on the bottom line. We can find 3 such triangles.
Pointing Down: These triangles have a base of length 2 on the top line. We can find 3 such triangles.
Total Size-2 Triangles = \(3 + 3 = 6\). 3. Triangles of Side Length 3:
Pointing Up: There is 1 such triangle, with its base on the bottom line.
Pointing Down: There is 1 such triangle, with its base on the top line.
Total Size-3 Triangles = \(1 + 1 = 2\).
So far, we have counted all the equilateral triangles: \(12 + 6 + 2 = 20\). To reach the answer of 28, we must find other types of triangles. 4. Non-Equilateral (Scalene/Isosceles) Triangles:
The grid lines also form larger, non-equilateral triangles. Let's identify them. These are typically "tilted" triangles spanning multiple grid units in height and width.
Consider the large triangles with a horizontal base of length 2 and whose third vertex is two units higher or lower. For example, a triangle with vertices at the bottom-left corner, the third bottom vertex, and a vertex on the top row.
A systematic search reveals 8 such triangles:
4 triangles are "pointing" towards the right.
4 triangles are "pointing" towards the left (as mirror images of the first 4).
Total Non-Equilateral Triangles = 8.
Step 3: Final Answer:
Summing up all the types of triangles we have counted:
\[ \text{Total Triangles} = (\text{Size-1}) + (\text{Size-2}) + (\text{Size-3}) + (\text{Non-Equilateral}) \]
\[ \text{Total Triangles} = 12 + 6 + 2 + 8 = 28 \]
Thus, there are 28 triangles in total in the figure.
