Chapter 5 – Introduction To Euclid’s Geometry Exercise 5.1

Que.1 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 .

Ans.

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.

Solution:-

False

Infinite numbers of line can be drawn through the single point

If we take point D on note-book and draw a line with the help of scale and pencil, we can draw an infinite numbers of line throught that point.

(ii) There are an infinite number of lines which pass through two distinct points.

Solution:-

False

We can draw infinite number of lines through point D. But can draw only single unique line from point D to point E.

(iii) A terminated line can be produced indefinitely on both the sides.

Solution:-

True

As per Euclid’s postulate 2 , A terminated line can be produced indefinitely.

(iv) If two circles are equal, then their radii are equal.

Solution:-

True

If two circle are equal then their centre and circumference will coincide.So, the radius is also equal.

(v) In fig. 5.9, If  AB=PQ and   PQ=XY, then  AB=XY .

Solution:-

True

Things which are equal to the same thing are equal to one another.(Euclid’s axiom)

Que. 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?

(i) parallel lines

(ii)perpendicular lines

(iii)line segment

(iv)radius of a circle

(v)Square

Ans.

(i)Parallel lines – Those straight lines which have no common point or which never intersect each other and are always at same distance.

(ii)Perpendicular lines – Those straight lines which intersect each other in apalen at right anglee then the lines are said to be perpendicular to each other.

(iii) line segment-A line with two end points is called a line-segment

(iv)Radius of a circle – the distance between the centre and circumfernce is known as radius of circle.

(v)Square- A quadrilateral having four equal sides and each angle is right angle, is called square.

Other Terms:-

Point – A point has no part.

Line- A line is a bredthless length. It can draw from one point to another point.

Straight Line– A line having no curves.

Angle– an angle is the figure formed by two straight rays.

Right angle– An angle which is equal to  900

Quadrilateral- A quadrilateral is four-sided plane figure and having four angle. Circle– The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

Circumference of circle– The length of the complete circle is called its circumference.       

Centre of circle– A fixed point inside the circle which is at the same distance from all points on the circle.

Que. 3 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.

Ans

Yes. These postulates contain two undefined terms ‘point’ and ‘lines’.

Yes. These postulates are consistent. Both the postulates describe two different facts.

No, they do not follow Euclid’s postulates.

In postulate (i) ‘in between A and B’ is not clear which appeals to geometric intuition.

Que.4 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

Ans

Given:-

AC = BC

To Prove:-

AC = 1/2 AB.

Proof:-

AC = BC (given)

By adding AC in both the sides,

∵ AC = BC and If equals are added to equals, the wholes are equal.

⇒ AC + AC = BC +  AC

∵ AB coincides with AC + BC.

⇒ 2 AC = AB

∵ Things which are halves of the same things are equal to one another.

⇒AC = 1/2 AB proved

Que (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.

Ans

Given:-

C is a mid-point of line segment AB

To prove:-

Every line segment has one and only one mid-point

Proof:-

Suppose that C andर D  are two mid-point of line segment AB .

C is a mid-point of line segment AB (given)

 AC = 1/2 AB.

AD = 1/2 AB.

⇒ AC = AD  (∵Things which are equal to the same thing are equal to one another)

But it is possible only if D coincides with C .

 C is the unique mid-point

Hence, every line segment has one and only one mid-point. Proved

Que. 6. In Fig. 5.10, if AC = BD, then prove that AB = CD.

Ans

Given:-

AC = BD

To Prove:-

AB = CD

Proof:-

AC = BD (given)

By substracting BC from both the sides,

∵ AC = BD and If equals are subtracted from equals, the reminders are equal.

⇒ AC – BC = BD –  BC

⇒ AB = CD

⇒AB=CD proved

Que.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.

Ans.

Axiom 5 the whole is greater than the part.

It is true for all fields. So, it is a universal truth.

 Example (i):-

9 = 6+3

6 and 3 is tha part of number 9   

9 > 6

9 > 3

 Example (ii):-

Chandigarh is a part of India.

India is greater that Chandigarh

So, the whole is greater than the part.

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