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Q4:
If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$. Explain by drawing the figure.
Solution :
Given Variables & Initial Setup
Let $A$, $B$, and $C$ be three points on a straight line such that $C$ lies between $A$ and $B$.
- The position of points is defined as: $A - C - B$ [By order of points on a line].
- It is given that $AC = BC$ [Given Condition].
Step 1: Establishing the Segment Addition Axiom
According to Euclid’s Common Notion 4: "Things which coincide with one another are equal to one another." In the context of line segments, if $C$ lies between $A$ and $B$, the sum of the parts equals the whole:
$AC + BC = AB$ [By Euclid's Axiom: The whole is greater than the part, and specifically, if a point $C$ lies between $A$ and $B$, then $AC + CB = AB$].
Step 2: Algebraic Substitution
Since we are given $AC = BC$, we can substitute $AC$ for $BC$ in the equation established in Step 1:
$AC + AC = AB$ [Substitution Property of Equality].
$2AC = AB$ [By arithmetic summation].
Step 3: Derivation of the Final Identity
To isolate $AC$, we divide both sides of the equation by 2:
$\frac{2AC}{2} = \frac{AB}{2}$ [By Euclid’s Common Notion 3: If equals are subtracted from equals, the remainders are equal; here applying the division axiom].
$AC = \frac{1}{2}AB$.
Visual Representation
The figure illustrates three collinear points where the distance $AC$ is equal to the distance $BC$. Since the total distance $AB$ is the sum of $AC$ and $BC$, and $AC = BC$, it is mathematically necessary that $AC$ is exactly one-half of the total length $AB$.
Final Solution: $AC = \frac{1}{2}AB$
More Questions from Class 9 Mathematics Introduction to Euclid’s Geometry EXERCISE 5.1
- Q1(i): 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.
- Q1(ii): Which of the following statements are true and which are false? Give reasons for your answers. (ii) There are an infinite number of lines which pass through two distinct points.
- Q1(iii): Which of the following statements are true and which are false? Give reasons for your answers. (iii) A terminated line can be produced indefinitely on both the sides.
- Q1(iv): Which of the following statements are true and which are false? Give reasons for your answers. (iv) If two circles are equal, then their radii are equal.
- Q1(v): Which of the following statements are true and which are false? Give reasons for your answers. (v) In Fig. 5.9, if $AB = PQ$ and $PQ = XY$, then $AB = X
- Q2(i): 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 lines
- Q2(ii): 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? (ii) perpendicular lines
- Q2(iii): 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? (iii) line segment
- Q2(iv): 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? (iv) radius of a circle
- Q2(v): 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? (v) square
- Q3: 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.
- Q5: 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.
- Q6: In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
- Q7: Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
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