Question

"Question 5 In the above question, point C is called a mid-point of line segment AB, prove that every line segment has one and only one mid-point.

Class 9 - Math - Introduction to Euclid's Geometry Page 86"

Answer 1

Firstly, let there are two midpoints of line segment AB and then use axiom 1, which states that which are equal to the same things are equal to one another.

For e g
AB=CD & CD= EF then AB=EF

i.e, If a line segment AB is equal to line segment CD and line segment CD is equal to line segment EF then line segment AB is equal to line segment EF.
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[Fig. Is in the attachment]

Solution:
We have C as a midpoint of line segment AB , so AC=BC.

Let there are two midpoints C & C' of AB.

Then , AC=1/2AB & AC'=1/2AB

AC= AC' (by axiom 1)

Which is possible only when C & C' coincide.
So ,point C' lies on C.

Hence, every line segment has one and only one midpoint.


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Hope this will help you....

Answer 2

Hi Friend..

A__________*D____C*_________B

Solution---If possible,let D be another mid-point of AB ..............equ--((1))

AD=DB

But it is given that C is the mid point of AB ...............equ--((2))

AC=CB

Subtracting (1) from(2)
Ac -Ad=cb-db
Dc=-DC
2DC =0
DC=0

∆∆∆∆∆∆So, C And D coincide.

Thus, every line segment had one and only one mid point.

Hence, proved...

Hope it is helpful...