# 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.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 5

**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.

**☛ Related Questions:**

- Which of the following statements are true and which are false? Give reasons for your answers.i) Only one line can pass through a single point.ii) There are an infinite number of lines which pass through two distinct points.iii) A terminated line can be produced indefinitely on both the sides.iv) If two circles are equal, then their radii are equal.v) In fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
- Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?i) Parallel linesii) Perpendicular linesiii) Line Segmentiv) Radius of a circlev) Square
- Consider two 'postulates' given below:i) Given any two distinct points A and B there exists a third point C which is in between A and B.ii) There exist at least three points that are not on the same line.Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.
- If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2 AB. Explain by drawing the figure.

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