Question

# Prove that every line segment has one and only one midpoint.

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Hint: Midpoint: To prove this given condition we assume that a line has two different mid-points. Then we apply the condition of Midpoint on them. If there is contradiction in the answer we get then our assumption will be wrong that a line has two midpoints. Through this assumption we prove our question.

Midpoint: A point between two points such that all three points are collinear, divides the line in such a way that distance from both the end points to that point is the same as Midpoint.

Let us consider, a line segment ${\text{AB}}$

Assume that it has two midpoints, say ${\text{C}}$ and ${\text{D}}$.

The midpoint of a line segment divides it into two equal parts.

That is,

${\text{AC = CB eq}}{\text{.1}} \\ {\text{and AD = DB eq}}{\text{.2 }} \\$

Since C is midpoint of ${\text{AB}}$ line segment,

we have ${\text{A, C}}$ and ${\text{B}}$ are collinear

${\text{AC + CB = AB eq}}{\text{.3}}$

Similarly, for midpoint ${\text{D}}$ of ${\text{AB}}$ line segment,

${\text{A, D}}$ and ${\text{B}}$ are collinear

${\text{AD + DB = AB eq}}{\text{.4}}$

From eq.3 and eq.4, we get

$\Rightarrow {\text{AC + CB = AD + DB}} \\$

From eq.1 and eq.2 we can rewrite the above equation as
$\Rightarrow {\text{ 2AC = 2AD}} \\ \Rightarrow {\text{ AC = AD}} \\$

This is a contradiction unless ${\text{C and D}}$ coincide.

Therefore, our assumption that a line segment AB has two midpoints is incorrect.

Hence prove that every line segment has one and only one midpoint.

Note: Whenever you get this type of question the key concept of solving the problem is that first you have the knowledge about the concept of question like in a problem you have to have knowledge about midpoint. Then, you have to assume the alleged condition and apply the concepts to prove this if there will be a contradiction in the answer you get after solving this it means your assumption is wrong . Hence our standard condition of question is true.