In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.
Let A and B be the line segment and points P and Q be two different mid points of AB.
Now,
∴ P and Q are midpoints of AB.
Therefore AP=PB and also AQ = QB.
also, PB + AP = AB (as it coincides with line segment AB)
Similarly, QB + AQ = AB.
Now,
AP + AP = PB + AP (If equals are added to equals, the wholes are equal.)
⇒ 2 AP = AB --- (i)
Similarly,
2 AQ = AB --- (ii)
From (i) and (ii)
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, P and Q are the same points. This contradicts the fact that P and Q are two different mid points of AB. Thus, it is proved hat every line segment has one and only one mid-point.
