Theorem:
Statement:
If a point \(W\) lies between two points \(U\) and \(V\) such that \(UW =VW\), then prove that \(UW = \frac{1}{2} UV\). Here, point \(W\) is called a mid-point of line segment \(UV\). Prove that every line segment has one and only one mid-point.
Proof:
Let there be two mid-points, \(W\) and \(F\).
\(W\) is the mid-point of \(UV\).
\(UW =WV\),
\(UW +UW =WV+UW \) () … (1)
Here, \((VW + UW)\) coincides with \( UV\).
It is known that by things which coincide with one another are equal to one another.
Therefore, \(VW + UW = UV\) … (2)
It is also known that by things which are equal to the same thing are equal to one another.
Therefore, from equations (1) and (2), we obtain
\(UW + UW = UV\)
That is, \(2UW = UV\) … (3)
Similarly, by taking \(F\) as the mid-point of \( UV\), it can be proved that \(2UF = UV\) … (4)
From equation (3) and (4), we obtain
\(2UW = 2UF\)
Therefore, \(UW = UF\)
This is possible only when point \(W\) and \(F\) are representing a single point.
Hence, our assumption is wrong and there can be only one mid-point of a given line segment.