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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.Q.5

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This is very important question of ncert class 9th of chapter Introduction To Euclid’s Geometry  . How I solve the best solution of exercise 5.1 question number 5. Please help me to solve this in a easy and best way.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.

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  1. Solution:

    Ncert solutions class 9 chapter 5-3

    Let, AB be the line segment

    Assume that points P and Q are the two different mid points of AB.

    Now,

    ∴ P and Q are midpoints of AB.

    Therefore,

    AP = PB and AQ = QB.

    also,

    PB+AP = AB (as it coincides with line segment AB)

    Similarly, QB+AQ = AB.

    Now,

    Adding AP to the L.H.S and R.H.S of the equation AP = PB

    We get, AP+AP = PB+AP (If equals are added to equals, the wholes are equal.)

    ⇒ 2AP = AB — (i)

    Similarly,

    2 AQ = AB — (ii)

    From (i) and (ii), Since R.H.S are same, we equate the L.H.S

    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, we conclude that P and Q are the same points.

    This contradicts our assumption that P and Q are two different mid points of AB.

    Thus, it is proved that every line segment has one and only one mid-point.

    Hence Proved

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