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Q6:
In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
Solution :
Step 1: Identifying Given Parameters and Geometric Context
Given the linear arrangement of points $A, B, C,$ and $D$ on a single line segment as shown in the geometric configuration:
- $AC = BD$
- Points $A, B, C, D$ are collinear and ordered such that $B$ lies between $A$ and $C$, and $C$ lies between $B$ and $D$.
[Per Euclid’s Axioms, specifically the segment addition postulate, the length of a segment composed of adjacent parts is the sum of those parts.]
Step 2: Formulating the Algebraic Equations
Based on the relative positions of the points, we express segments $AC$ and $BD$ as the sums of their constituent segments:
$AC = AB + BC$ [Equation 1]
$BD = BC + CD$ [Equation 2]
Step 3: Applying Euclid's Axioms
We are given the equality $AC = BD$. By substituting the expressions from Equation 1 and Equation 2 into this equality, we obtain:
$AB + BC = BC + CD$
Step 4: Algebraic Reduction
To isolate $AB$ and $CD$, we apply Euclid’s Third Common Notion: "If equals be subtracted from equals, the remainders are equal."
Subtracting the segment $BC$ from both sides of the equation:
$(AB + BC) - BC = (BC + CD) - BC$
$AB = CD$
Step 5: Theoretical Justification
The derivation holds because $BC$ is a common length present in both original segments $AC$ and $BD$. By removing the overlapping segment, the remaining non-overlapping parts $AB$ and $CD$ must necessarily be of equal magnitude.
Final Solution: $AB = CD$ (Proved)
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.
- 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.
- 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.
- 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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