Generating a specific graphic with TikZ

Question

I am wondering how to generate a specific image with TikZ. This image is essentially a lattice grid without the grid, with a triangle, rectangle, and other shapes overlayed in a particular way. I would also like to be able to notate points on specific lattice points. Here is an example of what I would like to create:

Thank you so much!

Answer 1

The dotted grid can be made with with the \draw command with grid but use line cap=round, dash pattern=on 0pt off 1cm (assuming you want 1cm grid spacing). Then the triangle can be made with a \draw command and labeled at the vertices using node.

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\draw[line width=2pt, line cap=round, dash pattern=on 0pt off 1cm](0,0) grid (9,6);
\draw (0,0)node[below]{$(0,0)$}--(8.5,0)node[below]{$(\frac{p}{2},0)$}--(8.5,5.5)node[above]{$(\frac{p}{2},\frac{q}{2})$}--cycle;
\end{tikzpicture}

\end{document}

Answer 2

As @Miyase said, you can found all what you need to draw a TiKz picture on CTAN. This answer is only for a starting point for you.

\documentclass[margin=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning, calc}
\tikzset{circle node/.style = {circle,inner sep=1pt,draw, fill=white}        
}

\begin{document}
  \begin{tikzpicture}[scale=0.5]
    \foreach \x in {0,...,9}
    \foreach \y in {0,...,6}
    {
    \fill (\x,\y) circle (1pt);
    }
    \foreach \y in {0,...,6}
    {
    \fill (10,\y) circle (1pt);
    }
    \foreach \x in {0,...,8}
    {
    \fill (\x,7) circle (1pt);
    }
    \draw (0,0)--(9.5,0)--(9.5,6.5)--cycle;
    \node at (0,0)[below]{$(0,0)$};
    \node at (9.5,0)[below]{$(\frac{p}{2},0)$};
    \node at (9.5,6.5)[above]{$(\frac{p}{2},\frac{q}{2})$};
  \end{tikzpicture}
\end{document}