NCERT Solutions Class 9 Maths Chapter 5- Introduction to Euclids Geometry

Introduction to Euclid’s Geometry

Exercise 1

Q1. 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 Figure, if AB = PQ and PQ = XY, then AB = XY.

Answer: 

1. A single point can have an endless number of lines traced through it. As a result, the statement in question is False.

2. There is only one line that may be drawn between two different points. As a result, the statement in question is False.

3. A terminated line can be created endlessly on both sides, just as a line can be stretched infinitely on both sides. As a result, the assertion is correct.

4. When the radii of two circles are the same, they are equal. Because the circumference and center of both circles are the same, the radius of the two circles should be the same. As a result, the statement is correct.

5. “Things that are equivalent to the same thing are likewise equal to one another,” says Euclid’s first axiom. As a result, the assertion is correct.

Q2. 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.  radius of a circle

5. Square

Answer: Yes, there are several concepts that must be defined beforehand, including:

1. Plane – Plane refers to flat surfaces on which geometric shapes can be drawn. A level surface is one on which the straight lines are uniformly distributed.

2. Point- A dimensionless dot painted on a planar surface is referred to as a point. A point is anything that does not belong.

3. Line- A line is a group of points that have the same length but no width. It may also be stretched in both directions. A line is a length without width.

1. Parallel lines are ones that never intersect and are always perpendicular to one another. There could be two or even more parallel lines.

2. Perpendicular lines – Perpendicular lines are those lines that meet in a plane at right angles and are thus perpendicular to each other.

3. Line Segment – A line segment is formed when a line could not be stretched any further due to its two end points. There are two end points to a line segment.

4. Radius of a circle – The radius of a circle is the line that connects any point on its perimeter to its center.

5. Square- A quadrilateral with all four sides equal and each interior angle at right angles is termed 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.

Answer: Yes, there are undefined words in these postulates. The following are undefined terms in the postulates:

In a plane, there are several points. However, the position of the point C, whether it is on the line segment connecting AB or not, is not specified in the postulates.

Furthermore, there is no indication if the points are in the same plane.

Yes, these postulates are consistent when applied to the following two scenarios: -Between A and B, point C is on the line segment AB.

-The line segment AB does not pass-through point C.

No, Euclid’s postulates do not imply them. They adhere to the axioms.

Q4. If a point C lies between two points A and B such that AC = BC, then prove that AC = ½ AB. Explain by drawing the figure.

Answer: 

We are given that, AC=BC

Now, we will be adding AC to both the sides of the equation such that:

AC+AC=BC+AC\\ 2AC=BC+AC\\

We already know that,

BC+AC=AB\\ => 2AC=AB\\ => AC=\frac{1}{2}AB

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.

Answer:

 Let us assume AB be the line segment

Also, points P and Q are the two separate middle points of the line segment AB.

Now we can write: AP=PB and AQ=QB.

Another way: PB+AP=AB  and QB+AQ=AB.

We will be adding AP to both the sides of the equation AP=PB.

AP+AP=PB+AP\\ => 2AP=AB[PB+AP=AB] –(i)

Now, we will add QB to both the sides of the equation AQ=QB.

AQ+AQ=QB+AQ\\ => 2AQ= AB[QB+AQ=AB] –(ii)

In equations (i) and (ii) we equal the L.H.S since the R.H.S are the same.

2 AP = 2 AQ (Things that are equal to one another are the same thing.)

⇒ AP = AQ (Things that are doubles of one another are equal.)

As a result, we deduce that P and Q are just the same point.

This contradicts our belief that P and Q are two distinct AB midpoints.

As a result, it is demonstrated that each line segment has just one mid-point.

Q6. In Fig. 5.10, if AC = BD, then prove that AB = CD.

Answer: 

We are given the information that AC=BD.

From the given diagram, we can deduce: 

AC=AB+BC

BD=BC+CD

Since AC=BD (given)

AB+BC=BC+CD

We know that when equals are subtracted from equals, the remainders are likewise equal, according to Euclid’s axiom.

Subtracting BC from both the sides of the above equation.

AB=CD

Therefore, we proved 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.)

Answer: Axiom 5 states that the whole is always larger than the sum of its parts.

Consider the case of a cake. Assume it weighs 2 pounds when entire or complete, but when a piece of it is removed and measured, it will weigh less than the prior measurement. As a result, Euclid’s fifth axiom holds true for all materials in the universe. As a result, Axiom 5 of Euclid’s axioms is regarded as a “universal truth.”

Exercise 5.2

Q1. How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?

Answer: The fifth postulate of Euclid states that if a straight line falls on two straight lines, the interior angles on the same side are less than two right angles when added together, then the two straight lines will meet on the side where the sum of angles is less than two right angles if produced indefinitely.

Q2. Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

Answer: The presence of parallel lines is implied by Euclid’s fifth postulate.

If the total of the interior angles equals the total of the right angles, the two lines will never intersect, making them parallel.