Question

point o is between points m and n on line segment mn. o is the midpoint of line segment mn if ______. options: on = nm, on + om = mn, om = nm, on = om

Explanation:

1. Recall the definition of a midpoint of a line segment: A midpoint of a line segment is a point that divides the segment into two segments of equal length.
2. Analyze each option:

• Option \( ON = NM \): \( NM\) is the same as \( MN\) (length of the whole segment), and \( ON\) is a part of \( MN\) (if \( O\) is the midpoint, \( ON=\frac{1}{2}MN\)), so this is false.
• Option \( ON + OM = MN\): This is true for any point \( O\) on segment \( MN\) (by the segment addition postulate), not just the midpoint.
• Option \( OM = NM\): \( NM = MN\) (length of the whole segment), and \( OM\) is a part of \( MN\) (if \( O\) is the midpoint, \( OM=\frac{1}{2}MN\)), so this is false.
• Option \( ON = OM\): This satisfies the definition of a midpoint (dividing \( MN\) into two equal - length segments \( OM\) and \( ON\)).

Answer:

B. \( OM = NM \) (Wait, actually the correct one should be \( OM = ON \) or \( OM = MN/2 \), but from the options, let's re - evaluate. The midpoint divides the segment into two equal parts. So if \( O \) is the midpoint of \( MN \), then \( OM = ON \) and \( OM+ON = MN \). But looking at the options:

Option 1: \( ON = NM \) → No, \( NM = MN \), \( ON \) is half of \( MN \) (if midpoint), so this is wrong.

Option 2: \( ON + OM=MN \) → This is always true if \( O \) is on \( MN \), not just when it's the midpoint.

Option 3: \( OM = NM \) → \( NM = MN \), \( OM \) is half of \( MN \) (if midpoint), so wrong.

Option 4: \( ON = OM \) → This is the definition of midpoint: the midpoint of a segment is a point that divides the segment into two congruent (equal - length) segments. So \( O \) is the midpoint of \( MN \) if \( ON = OM \). So the correct option is the one with \( ON = OM \) (the fourth option in the original, but let's check the labels. The options are:

1. \( ON = NM \)

1. \( ON + OM = MN \)

1. \( OM = NM \)

1. \( ON = OM \)

So the correct answer is the fourth option: \( ON = OM \) (the option labeled as \( ON = OM \), which is the last option in the vertical list).