A figure consists of a square divided into 9 equal smaller squares ($3\times3$), and 2 diagonals are drawn in the outer square. How many triangles are there?

Question

Prepp Team · Accepted Answer

Understanding the Figure

The problem describes a geometric figure based on a square. This square is divided into 9 equal smaller squares, forming a grid pattern that is 3 squares wide and 3 squares tall (a $3 \times 3$ grid). Additionally, the two main diagonals of the large outer square are drawn. We need to find the total number of triangles present in this composite figure.

Strategy for Counting Triangles

To count all the triangles accurately, we need a systematic approach. We can break down the counting process into categories, primarily by the size or the number of basic regions each triangle is composed of. The lines involved are the horizontal grid lines, the vertical grid lines, and the two diagonals of the outer square.

Counting Smallest Triangles

First, let's identify the smallest triangles formed by the intersecting lines. These are triangles that cannot be further subdivided by any line within the figure.

Center Square: The diagonals intersect in the center, dividing the central $1 \times 1$ square into 4 small triangles.
Corner Squares: There are 4 corner squares in the $3 \times 3$ grid. Each diagonal line passes through two of these corner squares. Each corner square is divided into 2 triangles by the diagonal passing through it. So, $4 \text{ corner squares} \times 2 \text{ triangles/square} = 8 \text{ triangles}$.
Edge Squares: There are 4 squares located in the middle of the outer edges of the $3 \times 3$ grid. The diagonals also pass through these squares. Similar to the corner squares, each of these 4 edge squares is divided into 2 triangles by the diagonal. So, $4 \text{ edge squares} \times 2 \text{ triangles/square} = 8 \text{ triangles}$.

The total number of these smallest, indivisible triangles is $4 + 8 + 8 = 20$.

Counting Medium Triangles

Next, we look for triangles formed by combining two adjacent smallest triangles. These triangles will be larger than the smallest ones but still relatively simple in structure. A key set of these triangles have their apex at the very center of the figure (where the diagonals intersect).

Consider the center point of the figure. Triangles can be formed using this center point as the apex and bases along the lines extending towards the outer edges. Specifically, consider triangles formed by:
- The center point.
- A vertex of the large square (e.g., the bottom-left corner).
- The midpoint of the outer edge adjacent to that vertex (e.g., the midpoint of the bottom edge).
An example of such a triangle has vertices at the center (1.5, 1.5), the bottom-left corner (0,0), and the midpoint of the bottom edge (1.5, 0). These triangles effectively combine one of the central triangles and a portion of an adjacent corner or edge triangle.
There are 8 such triangles identified around the center point. (4 triangles might be considered 'pointing' along the axes, and 4 along the diagonals, depending on orientation).

Counting these medium-sized triangles gives us 8 more.