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Question

# Decide whether each statement is always true, sometimes true, or never true. Justify your answers. a) Two line segments with the same midpoint have the same length. b) Two parallel line segments have the same midpoint. c) The midpoint of a line segment is the only point that divides it into two equal parts. d) A point equidistant from the endpoints of a line segment is the midpoint of the line segment.

Solution

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$\textbf{a)}$

Sometimes true: When two line segments of different length bisect each other, then they have the same midpoint.

$\textbf{b)}$

Never true: Two line segments only have the same midpoint if they intersect each other or are the same segment. Since parallel line segments do not intersect, they can never have the same midpoint.

$\textbf{c)}$

Always true: Only one point on the line segment can be equidistant to the endpoints, which is the midpoint.

$\textbf{d)}$

Sometimes true: It is the midpoint only if the point lies on the line segment itself. Otherwise, it is only a point on the right bisector of the line segment.

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