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Sagot :
To determine the condition under which point [tex]\( O \)[/tex] is the midpoint of the line segment [tex]\( MN \)[/tex], we can analyze each of the given options.
1. [tex]\( ON + OM = MN \)[/tex]
- This condition states that the sum of the distances from [tex]\( O \)[/tex] to [tex]\( N \)[/tex] and from [tex]\( O \)[/tex] to [tex]\( M \)[/tex] is equal to the total length of the segment [tex]\( MN \)[/tex].
Thus:
[tex]\[ ON + OM = MN \][/tex]
While this equation is true by the definition of a line segment, it does not specifically characterize [tex]\( O \)[/tex] as the midpoint. The requirement for [tex]\( O \)[/tex] to be the midpoint is that [tex]\( ON \)[/tex] and [tex]\( OM \)[/tex] be equal, which this equation alone does not imply.
2. [tex]\( ON = OM \)[/tex]
- This condition states that the distance from [tex]\( O \)[/tex] to [tex]\( N \)[/tex] is equal to the distance from [tex]\( O \)[/tex] to [tex]\( M \)[/tex].
Thus:
[tex]\[ ON = OM \][/tex]
This equality directly implies that [tex]\( O \)[/tex] is equidistant from both [tex]\( M \)[/tex] and [tex]\( N \)[/tex]. Therefore, [tex]\( O \)[/tex] would divide [tex]\( MN \)[/tex] into two equal segments, confirming that [tex]\( O \)[/tex] is the midpoint of [tex]\( MN \)[/tex].
3. [tex]\( ON = NM \)[/tex]
- This condition states that the distance from [tex]\( O \)[/tex] to [tex]\( N \)[/tex] is equal to the distance from [tex]\( N \)[/tex] to [tex]\( M \)[/tex].
Thus:
[tex]\[ ON = NM \][/tex]
This would imply that [tex]\( N \)[/tex] is the midpoint of the segment [tex]\( OM \)[/tex], which contradicts the definition of [tex]\( O \)[/tex] being the midpoint of [tex]\( MN \)[/tex]. Therefore, this is incorrect.
4. [tex]\( OM = NM \)[/tex]
- This condition states that the distance from [tex]\( O \)[/tex] to [tex]\( M \)[/tex] is equal to the distance from [tex]\( N \)[/tex] to [tex]\( M \)[/tex].
Thus:
[tex]\[ OM = NM \][/tex]
This would imply that [tex]\( M \)[/tex] is the midpoint, not [tex]\( O \)[/tex]. Therefore, this condition does not show that [tex]\( O \)[/tex] is the midpoint of [tex]\( MN \)[/tex].
Therefore, the correct condition that must hold true for [tex]\( O \)[/tex] to be the midpoint of [tex]\( MN \)[/tex] is:
[tex]\[ ON = OM \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
1. [tex]\( ON + OM = MN \)[/tex]
- This condition states that the sum of the distances from [tex]\( O \)[/tex] to [tex]\( N \)[/tex] and from [tex]\( O \)[/tex] to [tex]\( M \)[/tex] is equal to the total length of the segment [tex]\( MN \)[/tex].
Thus:
[tex]\[ ON + OM = MN \][/tex]
While this equation is true by the definition of a line segment, it does not specifically characterize [tex]\( O \)[/tex] as the midpoint. The requirement for [tex]\( O \)[/tex] to be the midpoint is that [tex]\( ON \)[/tex] and [tex]\( OM \)[/tex] be equal, which this equation alone does not imply.
2. [tex]\( ON = OM \)[/tex]
- This condition states that the distance from [tex]\( O \)[/tex] to [tex]\( N \)[/tex] is equal to the distance from [tex]\( O \)[/tex] to [tex]\( M \)[/tex].
Thus:
[tex]\[ ON = OM \][/tex]
This equality directly implies that [tex]\( O \)[/tex] is equidistant from both [tex]\( M \)[/tex] and [tex]\( N \)[/tex]. Therefore, [tex]\( O \)[/tex] would divide [tex]\( MN \)[/tex] into two equal segments, confirming that [tex]\( O \)[/tex] is the midpoint of [tex]\( MN \)[/tex].
3. [tex]\( ON = NM \)[/tex]
- This condition states that the distance from [tex]\( O \)[/tex] to [tex]\( N \)[/tex] is equal to the distance from [tex]\( N \)[/tex] to [tex]\( M \)[/tex].
Thus:
[tex]\[ ON = NM \][/tex]
This would imply that [tex]\( N \)[/tex] is the midpoint of the segment [tex]\( OM \)[/tex], which contradicts the definition of [tex]\( O \)[/tex] being the midpoint of [tex]\( MN \)[/tex]. Therefore, this is incorrect.
4. [tex]\( OM = NM \)[/tex]
- This condition states that the distance from [tex]\( O \)[/tex] to [tex]\( M \)[/tex] is equal to the distance from [tex]\( N \)[/tex] to [tex]\( M \)[/tex].
Thus:
[tex]\[ OM = NM \][/tex]
This would imply that [tex]\( M \)[/tex] is the midpoint, not [tex]\( O \)[/tex]. Therefore, this condition does not show that [tex]\( O \)[/tex] is the midpoint of [tex]\( MN \)[/tex].
Therefore, the correct condition that must hold true for [tex]\( O \)[/tex] to be the midpoint of [tex]\( MN \)[/tex] is:
[tex]\[ ON = OM \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
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