How many triangles in the diagram?

Question

I want to count the number of triangle in the following diagram. I have manually counted the no of triangles in the diagram. The no of triangles is 44.

Is there any other way to count the no of triangles?

Answer 1

There are various systematic ways to go about it. I would start by counting the ones that use the lower left to upper right diagonal and are below it. I find $4$ that are $1 \times 1$, $3$ that are $2 \times 2$, $2$ that are $3 \times 3$, and $1$ that is $4 \times 4$ for a total of $10$. There are $4$ symmetric locations for this and that exhausts the triangles with two sides parallel to the sides of the rectangle. There are also two triangles in the vee below the diagonals times four positions for $8$, giving a total of $48$.

Answer 2

A way of thinking of a triangle is the lines that make it up. In this case, you have 5 horizontal lines, 5 vertical lines and two diagonal lines for a total of 12. Now, to make a triangle no two of the lines can be parallel, and they cannot meet at a single point. Can you count them now using basic PnC?