What is the best way to explain counting triangles

Question

What is the best way to explain such type of questions to 11 years old? and also please let me know the answer as well. Thank you.

Answer 1

There is a natural assumption that the sides of the triangles have to be made up of the solid line segments in the diagram.

You need to find a systematic way of counting.

First note that every triangle you will be counting has three sixty degree angles and will therefore be equilateral.

Second, note that all such equilateral triangles will have precisely one side which is horizontal.

Third, each horizontal line can be part of at most two triangles - one above it and one below it.

So first look at horizontal segments of length $1$. The two at the top and the one at the bottom are part of just one triangle of side $1$ and the others count for two.

Then look at the horizontal line segments of length $2$ in a similar way. And then longer sides until you are done.

(note the side $1$ triangles are easy to count anyway, but it is sometimes useful to have a single systematic approach, rather than different approaches for different sizes - also you have an easy check on your count)