Question

if gregory draws the segment with endpoints b and c, then the midpoint will be on the line of reflection.
points a and b are equidistant from the line of reflection.
if gregory draws the segment with endpoints a and a, then the midpoint will lie on the line of reflection.
ab will be perpendicular to the line of reflection.

Explanation:

Step1: Recall the property of reflection

When a figure is reflected over a line, the line of reflection is the perpendicular bisector of the segment joining a point and its image.

Step2: Analyze the first statement

For a segment with endpoints \(B\) and \(C'\) (image of \(C\) after reflection), the mid - point of \(BC'\) lies on the line of reflection.

Step3: Analyze the second statement

Points \(A\) and \(B\) are equidistant from the line of reflection. But this is not the key property related to the mid - point and reflection.

Step4: Analyze the third statement

For a segment with endpoints \(A\) and \(A'\) (image of \(A\) after reflection), the mid - point of \(AA'\) lies on the line of reflection.

Step5: Analyze the fourth statement

\(\overline{AB}\) is not necessarily perpendicular to the line of reflection. The property is about the perpendicular bisector of the segment joining a point and its image.

Answer:

If Gregory draws the segment with endpoints \(B\) and \(C'\), then the midpoint will be on the line of reflection.