Question

How many different non-congruent isosceles triangles can be formed by connecting three of the dots in a 4\times4 square array of dots like the one shown below?

Answer 1

@ganeshie8

Answer 2

An array like this?\[\begin{matrix}
\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot
\end{matrix}\]

Answer 3

Let's start by fixing a vertex in (1,1):
\[\begin{matrix}
\color{red}\bullet&\rightarrow&\color{red}\bullet&&\cdot&&\cdot\\
\uparrow&\swarrow\\
\color{red}\bullet&&\cdot&&\cdot&&\cdot\\
\\
\cdot&&\cdot&&\cdot&&\cdot\\
\\
\cdot&&\cdot&&\cdot&&\cdot
\end{matrix}\]
\[\begin{matrix}
\color{red}\bullet&\rightarrow&\bullet&\rightarrow&\color{red}\bullet&&\cdot\\
\uparrow&&&\swarrow\\
\bullet&&\bullet&&\cdot&&\cdot\\
\uparrow&\swarrow\\
\color{red}\bullet&&\cdot&&\cdot&&\cdot\\
\\
\cdot&&\cdot&&\cdot&&\cdot
\end{matrix}\]
\[\begin{matrix}
\color{red}\bullet&\rightarrow&\bullet&\rightarrow&\bullet&\rightarrow&\color{red}\bullet\\
\uparrow&&&&&\swarrow\\
\bullet&&\cdot&&\bullet&&\cdot\\
\uparrow&&&\swarrow\\
\bullet&&\bullet&&\cdot&&\cdot\\
\uparrow&\swarrow\\
\color{red}\bullet&&\cdot&&\cdot&&\cdot
\end{matrix}\]
You thus have up to three 45-45-90 triangles for each vertex.

For a general counting strategy, you could try fixing two of the red vertices and counting how many vertices you can pick to make up an isosceles triangle.