What is a line segment called that connects 2 points on a circle and passes through the center?

Answer 1

First, just to perfect your knowledge, any line passing though two DISTINCT point on the circumference of a circle is called a SECANT LINE.

So we are discussing the special secant line that passes through the centre. It doesn't really have other names. What's special about it though, is the segment of this line that is contained between two points.

This line segment is what you are talking about, and the line itself is nothing big. But, the LENGTH of this line segment is defined as the DIAMETER of this circle.

Your teacher/friends might tell you that this LINE SEGMENT is the diameter, that is FALSE. Because there are infinitely many such lines can be drawn on a circle. But a circle have only ONE value for diameter.

Corollary:

Only the set of all such line segments have the length of the diameter of the circle, that's why these lines are special.