5. 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.

Answers (1)


Let's assume that there are two midpoints  C and D 
Now,
If C is the midpoint then,  AC = BC
And
In the figure given above, AB coincides with AC + BC.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB
From this, we can say that 
2AC = AB                                       -(i)

Similarly,
If D is the midpoint then, AD = BD
And
In the figure given above, AB coincides with AD + BD.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AD + BD = AB
From this, we can say that 
2AD = AB                                       -(ii)
Now,
From equation (i) and (ii) we will get 
AD = AC 
and this is only possible when C and D are the same points 
Hence, our assumption is wrong and there is only one midpoint  of line segment AB

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