In, point C is called a midpoint of line segment AB. Prove that every line segment has one and only one midpoint.
In, point C is called a midpoint of line segment AB. Prove that every line segment has one and only one midpoint.
Let, AB be the line segment
Assume that points P and Q are the two different mid points of AB.
∴ P and Q are midpoints of AB.
Therefore,
AP = PB and AQ = QB.
PB+AP = AB (as it coincides with line segment AB)
Similarly, QB+AQ = AB.
Adding AP to the L.H.S and R.H.S of the equation AP = PB
AP + AP = PB + AP (If equals are added to equals, the wholes are equal.)
⇒ 2AP = AB — (i)
Similarly,
2 AQ = AB — (ii)
From (i) and (ii), Since R.H.S are same, we equate the L.H.S
2 AP = 2 AQ (Things which are equal to the same thing are equal to one another.)
⇒ AP = AQ (Things which are double of the same things are equal to one another.)
Thus, we conclude that P and Q are the same points.
This contradicts our assumption that P and Q are two different mid points of AB.
Thus, it is proved that every line segment has one and only one midpoint.
Hence Proved.

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