How do you duplicate a path in tikz and rotate the duplicate about its midpoint?

I want to make a copy of the line inside the circle (imaged below with MWE) and draw it through the original line perpendicularly (or any angle I want for that matter).

Thank you very much for your assistance,
Pseudonym123

[![enter image description here](https://i.stack.imgur.com/F4udh.png)](https://i.stack.imgur.com/F4udh.png)

```
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}[scale=0.3]
\path[name path=circle,draw] (0,0) circle [radius=5cm];
\path[name path=line127] (3,0) -- ++(127:10);
\path[name intersections={of=circle and line127}]
    (intersection-1);
\path[save path=\name](3,0) -- (intersection-1);
\path[draw,rotate=90,use path=\name];%<--- this part isn't rotating
\end{tikzpicture}
\end{document}

```