GSEB Class 9 Maths Solutions Chapter 5 Introduction to Euclids Geometry Exercise 5.1

GSEB Solutions

Question 1. Which of the following statements are true and which are false? Give reasons for your answers.
1. Only one line can pass through a single point.
2. There is an infinite number of lines that pass through two distinct points.
3. A terminated line can be produced indefinitely on both sides.
4. If two circles are equal, then their radii are equal.
5. In figure, if AB = PQ and PQ = XY, then AB = XY.
Answer:
1. False. Through a single point, an endless number of lines can pass. In the following figure, we can clearly see that many lines can go through just one point.
2. False. Based on Euclid's first Postulate, only one specific line can be drawn through any two different points. The following image demonstrates that only a single line passes through two distinct points A and B.
3. True. Euclid's second Postulate says that a line segment can be made longer without end. A line segment is a part of a line, and it can stretch forever in two directions to become a full line.
4. True. If two circles are the same, they can be placed exactly on top of each other. This means their centers must line up, and their edges must match perfectly. Therefore, their radius measures must also be exactly the same.
5. True. This idea comes from Euclid's first common notion (Axiom 1), which tells us that if two different things are both equal to a third thing, then those first two things must also be equal to each other. Since AB and XY both equal PQ, then AB and XY have to be equal.
In simple words: For each statement, decide if it is correct or incorrect based on rules of geometry, then give the specific reason why. Remember key principles like how many lines pass through points or how line segments can be extended.

Exam Tip: Remember Euclid's five postulates and five common notions (axioms) as they form the foundation of geometry and are crucial for proving such statements.

Question 2. 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?
1. Parallel lines
2. Perpendicular lines
3. Line segment
4. The radius of a circle
5. Square
Answer:
1. Parallel lines: Those lines which do not cross each other are called parallel lines, or if the perpendicular distance between two lines stays constant, then these lines are said to be parallel. Example, \( AB \parallel CD \). To define parallel lines, we must know about points, lines, the distance between the lines, and the point of intersection.
2. Perpendicular lines: If two lines cross each other at 90°, then such lines are called perpendicular lines. Example: \( CD \perp AB \) or \( AB \perp CD \). Here we first need to define line and angle.
3. Line segment: A straight line segment drawn from any point to any other point. Before defining a line segment, we should know the point and straight line.
4. The radius of a circle: Distance between the center of a circle to any point lying on the circumference of the circle. Point and circle should be defined before defining the radius of a circle.
5. Square: A square is a quadrilateral that has all sides of equal length and all its inside angles are right angles, which means they are 90 degrees. To define a square, we must know about quadrilateral, side and angle.
In simple words: For each term like 'parallel lines' or 'square', state what it means and then list any simpler concepts you need to know first to understand that term, such as 'point' or 'line'.

Exam Tip: When defining geometric terms, always start with the most basic components (points, lines, planes) and build up to more complex figures, specifying properties like angles and lengths.

Question 3. Consider two 'postulates' given below:
1. Given any two distinct points A and B, there exists a third point C which is in between A and B.
2. 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.
Answer:
1. Yes, these postulates contain two undefined terms: point and line.
2. Yes, these postulates are consistent because they lead to two distinct situations:
(a) Given two points A and B, there exists a point C lying on the line between them.
(b) Given two points A and B, we can choose another point C which does not lie on the line that passes through A and B.
Therefore, these postulates do not follow from Euclid's postulates; however, they follow from Axiom 5.1.
In simple words: These rules use words like 'point' and 'line' which aren't fully defined. They work well together because they can describe two different true situations. They don't come directly from Euclid's original rules but are part of a later axiom.

Exam Tip: Postulates are fundamental statements assumed to be true without proof. It is important to identify any undefined terms and check for consistency when analyzing new postulates.

Question 4. 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.
Answer:
Given: \( AC = BC \)
We are given that \( AC \) equals \( BC \).
Add \( AC \) to both sides of the equation:
\( AC + AC = BC + AC \)
\( \implies 2AC = BC + AC \)
From the figure, C lies between A and B, which means that \( BC + AC \) represents the entire length of \( AB \).
So, we can write:
\( 2AC = AB \)
Now, divide both sides by 2:
\( AC = \frac{1}{2} AB \)
Thus, we have proven that if C is the midpoint of AB, then AC is half the length of AB.
In simple words: We are given that \( AC \) and \( BC \) are equal. If we add \( AC \) to both sides of this equality, we get two times \( AC \) on one side. On the other side, \( BC \) plus \( AC \) makes up the whole line \( AB \). So, two times \( AC \) equals \( AB \). Dividing by two shows that \( AC \) is half of \( AB \).

Exam Tip: For proofs, always start with what is given, use clear and logical steps, and refer to definitions or axioms where necessary. A clear diagram always helps in visualizing the problem.

Question 5. 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.
Answer:
Let's assume a line segment has two mid-points, C and D.
If C is the mid-point between A and B, then, as proved in Question 4:
\( AC = \frac{1}{2} AB \) .......(1)
If D is also a mid-point (as supposed), then similarly:
\( AD = \frac{1}{2} AB \) .......(2)
From equations (1) and (2), we can conclude that:
\( AC = AD \)
This outcome demonstrates that C and D must be the same point because their positions and distances from A are identical. Therefore, every line segment possesses one and only one mid-point.
In simple words: Imagine a line segment could have two middle points, C and D. If C is the middle, then \( AC \) is half the total length. If D is also the middle, then \( AD \) is also half the total length. This means \( AC \) and \( AD \) are the same, so C and D must be the same point, proving there's only one middle point.

Exam Tip: Proofs by contradiction (assuming the opposite and showing it leads to a contradiction) are a powerful tool in geometry. Clearly state your assumption and the logical steps that lead to the contradiction.

Question 6. In figure, if AC = BD, then prove that AB = CD
Answer:
Given: \( AC = BD \) ......(1)
From the figure, we observe that point B lies between A and C, so we can write:
\( AC = AB + BC \) .....(2)
Similarly, point C lies between B and D, so we can write:
\( BD = BC + CD \) .......(3)
Now, substitute the expressions for AC from (2) and BD from (3) into equation (1):
\( AB + BC = BC + CD \)
Now, subtract \( BC \) from both sides of the equation:
\( AB + BC - BC = BC + CD - BC \)
\( \implies AB = CD \)
This proves that \( AB \) is equal to \( CD \), which was required.
In simple words: We are given that the length \( AC \) is the same as \( BD \). From the picture, we know \( AC \) is made of \( AB \) plus \( BC \). We also know \( BD \) is made of \( BC \) plus \( CD \). If we put these into the first statement, we get \( AB + BC = BC + CD \). If we remove \( BC \) from both sides, we are left with \( AB = CD \).

Exam Tip: When dealing with line segments, make sure to correctly identify which segments are parts of longer ones (e.g., \( AC = AB + BC \)) and use substitution to simplify equations.

Question 7. 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).
Answer:
Axiom 5 (also known as Common Notion 5) states that "The whole is greater than the part." This statement is considered a 'universal truth' because its correctness applies to anything, anywhere in the world, without needing any proof. It holds true in all fields, not just geometry or mathematics. For example, a complete pizza is always larger than a single slice of that pizza, and a full country is always larger than a single state within it. These examples illustrate that the concept is inherently obvious and universally applicable. Therefore, it is a universal truth.
In simple words: Axiom 5 says that a whole thing is always bigger than just a part of it. This is true everywhere and for everything, like a whole cake is bigger than a slice. It doesn't need proof because it's always obvious and correct, making it a universal truth.

Exam Tip: Understand the difference between postulates (specific to geometry) and axioms/common notions (universal truths). Axioms are self-evident statements that apply across all branches of mathematics and logic.