Drawing and labeling the sides and angles of a right triangle

Question

The following code instructs TikZ to draw a right triangle AQP with a right angle at Q. I mark the length of line segment AQ by y. How do I rotate by 90 degrees the letter y so that it is upright? I would like to draw a line segment perpendicular to PA from A to the line containing the leg PQ, and I would like to label that point of intersection R.

\documentclass[10pt]{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}


\begin{document}



\begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt}]
\path
  (80:5) node [dot,label=above left:$A$]{} coordinate (A)
  (80:7) coordinate (a)
  (20:9) coordinate (B)
  (20:11) coordinate (b)
  (0:0) node[dot,label=below left:$P$]{} coordinate(P)
  (-100:1)coordinate (e)
  (-160:1) coordinate (f);

\path coordinate (Q)at($(P)!(A)!(B)$) node at (Q)[dot,label=below:$Q$]{} ;
 \draw[<->]  (a)--(e);
\draw[<->, name path=kline] (f)--(b) node[below right]{$k$}; % First line for intersection

\draw[purple!70!black,dashed] (A)--(Q);
\tkzMarkRightAngle(A,Q,P);

\draw ($(P)!3mm!90:(A)$)--($(A)!3mm!-90:(P)$)coordinate(u); % Note here the invisible point u, where uA is normal to pA at point A
\draw[|<->|] ($(P)!-7mm!90:(Q)$)--node[fill=white,sloped] {$x$} ($(Q)!-7mm!-90:(P)$);
\draw[|<->|] ($(Q)!-3mm!90:(A)$)--node[fill=white] {$y$} ($(A)!-3mm!-90:(Q)$);

\tkzMarkAngle[size=0.75cm,color=cyan,mark=||](B,P,A);
\tkzMarkAngle[size=1cm,color=cyan,mark=|](P,A,Q);

\path [name path=ARline] (u)--($(A)!-10cm!(u)$); % Second line for intersection
\path [name intersections={of = ARline and kline, by=R}];
\draw (A)--(R)node[dot,label=below:$R$]{};
\tkzMarkRightAngle(R,A,P);
\end{tikzpicture}

\end{document}

Answer 1

Your code produces errors; I tried to polish it by guessing what you were trying to do:

The code:

\documentclass[10pt]{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}


\begin{document}



\begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt}]
\path
  (80:5) node [dot,label=above left:$A$]{} coordinate (A)
  (80:7) coordinate (a)
  (20:9) node [dot,label=below:$B$]{} coordinate (B)
  (20:11) coordinate (b)
  (0:0) node[dot,label=below left:$P$]{} coordinate(P)
  (-100:1)coordinate (e)
  (-160:1) coordinate (f);

\node[dot,label=below:$Q$] (Q) at ($(P)!(A)!(B)$) {};
\node[dot,label=above right:$R$] (R) at ($(P)!(Q)!(A)$) {};

\draw[<->]  (a) -- (e);
\draw[<->]  (b) -- (f);

\draw[purple!70!black,dashed] (A) -- (Q);
\draw[green!70!black,dashed] (Q) -- (R);
\tkzMarkRightAngle(A,Q,P);

\draw[|<->|] ($(P)!3mm!90:(A)$)--node[fill=white,sloped] {$r$} ($(A)!3mm!-90:(P)$);
\draw[|<->|] ($(P)!-7mm!90:(Q)$)--node[fill=white,sloped] {$x$} ($(Q)!-7mm!-90:(P)$);
\draw[|<->|] ($(Q)!-3mm!90:(A)$)--node[fill=white] {$y$} ($(A)!-3mm!-90:(Q)$);

\tkzMarkAngle[size=0.75cm,color=cyan,mark=||](B,P,A);
\tkzMarkAngle[size=1cm,color=cyan,mark=|](P,A,Q);
\end{tikzpicture}

\end{document}

Answer 2

This is what is required in the OP last comments:

\documentclass[10pt]{standalone} 
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usepackage{tkz-euclide}
\usetkzobj{all}
\begin{document}

\begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt},line width=.7pt]
\path
  (80:5) node[dot,label=above left:$A$]{} coordinate (A)
  (80:7) coordinate (a)
  (20:9) coordinate (B)
  (20:11) coordinate (b)
  (0:0) node[dot,label=below left:$P$]{} coordinate(P)
  (-100:1)coordinate (e)
  (-160:1)coordinate (f)
  ($(P)!(A)!(B)$)node[dot,label=below:$Q$]{} coordinate(Q);

\path[name path=kline](f)--(b); % First line for intersection
\path[name path=ARline](A)--($(A)!10cm!90:(P)$); % A line normal to pA at point A in the east direction

\path [name intersections={of = ARline and kline, by=R}];
\draw (A)--(P)--(R)node[dot,label=below:$R$]{}--cycle;
\draw[purple!70!black,dashed] (A)--(Q);

\tkzMarkRightAngle(A,Q,P);
\tkzMarkAngle[size=0.75cm,color=cyan,mark=||](B,P,A);
\tkzMarkAngle[size=1cm,color=cyan,mark=|](P,A,Q);
\tkzMarkRightAngle(R,A,P);
\end{tikzpicture}
\end{document}

Which gives:

The best way to learn TikZ is to study others codes besides reading the comprehensive user manual (latest version).

Answer 3

Perhaps this is what you want?

\documentclass[10pt]{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}

\begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt}]

  \coordinate (Q) at (0,0);
  \coordinate (A) at ($(Q)+(30:3)$);
  \coordinate (P) at ($(Q)+(30+90:4)$);

  \draw (A) -- (Q) -- (P) -- cycle;

  \path (A) -- node [midway,sloped,above] {hypotenuse} (P);
  \path (A) -- node [midway,sloped,below] {shorter leg} (Q);
  \path (P) -- node [midway,sloped,below] {longer leg} (Q);

  \tkzMarkRightAngle(A,Q,P);
\end{tikzpicture}

\end{document}

You seem to be attempting to assign a lot of different coordinates, but I'm having a hard time seeing what you're trying to accomplish. But you asked about labeling the sides, so that's what I've done.