How to make a line intersect with a path at random angles?

How do you make a line intersect with the anchor point of a half circle when you want the line to continue on past the anchor point of the half circle at a random angle?

Howdy.
Try this:
Lock the semicircle. With the pen tool, draw a line from the point of origin to the anchor point of the semicircle. It will still snap even though it's locked.
Now, we use Jet's scale idea. Select the Scale Tool (S). Click on the point of origin to position the reference point. It will snap. Now, while holding down the Shift Key, drag right and down. The line segment will stay snapped to the endpoint while you drag (highlighted). You can drag it as far as you want.
That's it.
Peace,
Lee
PS: It's pretty much all Jet's method with the benefit of the OP's screenshot of what he was trying to accomplish. Jet offered the best answer, it looks like to me.

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