If a point C lies between two points A and B such that AC = BC, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.
Let there be two mid-points, C and D.
C is the mid-point of AB.
AC = CB
AC + AC = BC + AC (Equals are added on both sides) … (1)
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
∴ BC + AC = AB … (2)
It is also known that things which are equal to the same thing are equal to one another. Therefore, from equations (1) and (2), we obtain
AC + AC = AB
⇒ 2AC = AB … (3)
Similarly, by taking D as the mid-point of AB, it can be proved that
2AD = AB … (4)
From equation (3) and (4), we obtain
2AC = 2AD (Things which are equal to the same thing are equal to one another.)
⇒ AC = AD (Things which are double of the same things are equal to one another.)
This is possible only when point C and D are representing a single point.
Hence, our assumption is wrong and there can be only one mid-point of a given line segment.
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