# 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.

**Solution:**

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

Let us consider that line segment AB has two midpoints ‘C’ and ‘D’ as shown in the figure below.

Let's assume C to be the mid-point of AB

Thus, AC = BC

Adding AC on both sides, we get

⇒ AC + AC = BC + AC (BC + AC coincides to AB)

⇒ 2 AC = AB

⇒ AC = 1/2AB-----(1)

Let us consider a point D lying on AB,

Let's assume that D is another mid-point of AB.

Therefore AD = BD

Adding equal length AD on both sides, we get

AD + AD = BD + AD (BD + AD coincides to AB)

⇒ 2 AD = AB

⇒ AD = 1/2AB------(2)

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

- C has to coincide with D for AC to be equal to AD.
- According to Euclid's Axiom 4: Things which coincide with one another are equal to one another.
- Thus, a line segment has only one midpoint.

**Video Solution:**

## 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.

### NCERT Solutions Class 9 Maths - Chapter 5 Exercise 5.1 Question 5:

**Summary:**

Hence, using Euclid's axiom we proved that if point C is called a mid-point of line segment AB, then every line segment has one and only one mid-point.