# Ex.5.1 Q5 Introduction to Euclids Geometry Solution - NCERT Maths Class 9

## Question

In a 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.

## Text Solution

**Reasoning:**

We are aware that the things which coincide with one another are equal to one another.

**Steps:**

Let us consider that line segment \(AB\) has two mid points \(‘C’\) and \(‘D’\).

Let assume \( C\) is mid-point of \(AB\)

\[AC = BC\]

Adding \(AC\) on both sides, we get

\[\begin{align} AC+AC & = BC+AC \\ 2 AC & = AB \\ AC& = \frac { 1 } { 2 } AB \ldots \ldots ( 1 ) \end{align}\]

Let us consider a point \(D\) lying on \(AB,\) Let assume that \(D\) be another mid-point of \(AB.\)

Therefore \(AD = BD\) Adding equal length \(AD\) on both the sides, we get

\[\begin{align} AD+AD & = BD+AD \\ 2 AD & = AB \\ AD & = \frac { 1 } { 2 } AB \ldots \ldots ( 2 ) \end{align}\]

From equations (\(1\)) and (\(2\)), we can conclude that \(AC = AD\)

- \(C\) Coincides with \(D.\)
- Axiom \(4\): Things which coincide with one another are equal to one another.
- A line segment has only one midpoint.