Q. 54.2( 162 Votes )
In Question 4, po
Let A and B be the line segment and points P and Q be two different midpoints of AB.
AP = PB
AQ = QB
PB + AP = AB (It coincides with line segment AB)
QB + AQ = AB
AP + AP = AP + BP (Since, If equals are added to equals, the wholes are equal.)
2AP = AB .....(i)
2AQ = AB .....(ii)
From (i) and (ii),
2AP = 2AQ (Since things which are equal to the same thing are equal to one another)
And as we know:
Things which are double of the same thing are equal to one another.
AP = AQ
Thus, P and Q are the same points.
This contradicts the fact that P and Q are two different midpoints AB.
Thus, it is proved that every line segment has one and only one midpoint.
From the Figure,
AP + PB = AB.......................eq(i)
AQ + QB = AB.....................eq(ii)
From eq(i) and eq(ii)
AP + PB = AQ + QB
Now let AP = PB, and AQ = QB (as they are the midpoints, then)
2 AP = 2 AQ
AP = AQ
P = Q
Hence, there is only one midpoint.
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In the given figuRS Aggarwal & V Aggarwal - Mathematics