Systematic Approach to Counting Triangles

To accurately count all the triangles in the figure, we can break down the counting process by the size of the triangles, specifically by how many small, single components they are made of.

1. Triangles Made of 1 Component

First, we count the smallest, individual triangles. There are 4 in the top half of the figure and 4 in the bottom half.

Total = 4 (top) + 4 (bottom) = 8 triangles.

2. Triangles Made of 2 Components

Next, we find all triangles formed by combining two adjacent small triangles.

• In the top half, there are 3 such triangles.
• In the bottom half, there are 3 such triangles.
• Additionally, there are 2 larger triangles on the left and right sides (each pointing sideways), formed from two smaller components.

Total = 3 + 3 + 2 = 8 triangles.

3. Triangles Made of 3 Components

Now we look for triangles formed by combining three small components.

• In the top half, there are 2 such triangles.
• In the bottom half, there are 2 such triangles.

Total = 2 + 2 = 4 triangles.

4. Triangles Made of 4 Components

Finally, we count the largest triangles, which are each made of four small components.

• The entire top half forms one large triangle.
• The entire bottom half forms one large triangle.
• The entire left half forms one large triangle.
• The entire right half forms one large triangle.

Total = 4 triangles.

Total Calculation

To find the total number of triangles, we sum the counts from each category:

Total = (1-component) + (2-component) + (3-component) + (4-component)

Total = 8 + 8 + 4 + 4 = 24

There are a total of 24 triangles in the figure.