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A line segment that has been cut exactly in half
A bisector cuts a line SEGMENT into two congruent line segments. A line has indefinite or infinite length.
If you cut a rectangle in half you wouldn't get a solid figure at all, since a rectangle is a plane figure. If you made a straight line cut you would get either a triangle or a quadrilateral of some variety depending on exactly how the cut was made.
It is called a frustum.
I believe that is called a chord. Yes, a chord is the straight line segment that crosses a circle, from one side of the circle to the other. The biggest possible chord is the diameter. The curved part of the circle, cut off by the chord [or chords], is the arc or the angle.