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.
solution :
(i) False
There are be infinite number of lines that can be drawn through a single point.
Hence, the statement is False.
(ii) False
Through two distinct points there can be only one line that can be drawn.
Hence, the statement is False.
(iii) True
A line that is terminated can be indefinitely produced on both sides as a line can be extended on both its sides infinitely.
Hence, the statement is True.
(iv) True
The radius of the two circles should be equal.
Hence, the statement is True.
(v) True
The first axiom of Euclid.
Hence, the statement is True.
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
solution :
Yes, we need to have an idea about the terms like point, line, rey, angle, plane, circle and quadrilateral Etc. before defining the required terms.
Definitions of the required terms are given below:
(i) Parallel Lines
Two lines ` \m\ ` and ` \n\ ` in a plane are said to be parallel, if they have no common point and we write them as ` \m\ ` || ` \n\ `.
(ii) perpendicular Lines
Two lines ` \a\ ` and ` \b\ ` lying in the same plane are said to be perpendicular, if they form a right angle and we write4 them as ` \a\ ` ⊥ ` \b\ `.
(iii) Line Segment
A line sagment is a part of line and having a definite length. it has two end-points. a line segment is shown having end points A and B. It is written as `\overline{AB}\ ` or ` \overline{BA}\ `.
(iv) Radius of a circle
The distance from the centre to a point on the circle is called the radius of the circle. In the figure, P is centre and Q is a point on the circle, then PQ is the radius.
(v) Square
A quadrilateral in which all the four angles are right angles and all the four sides are equal is called a square. PQRS is a square.
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.
solution :
Yes, these postulates contain undefined terms such as ‘Point and Line’. Also, these postulates are consistent because they deal with two different situations as
(i) says that given two points A and B, there is a point C lying on the line in between them.
(ii) says that, given points A and B, you can take point C not lying on the line through A and B.
No, these postulates do not follow from Euclid’s postulates, however they follow from the axiom, “Given two distinct points, there is a unique line that passes through them.”
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.
solution :
Given that,
AC = BC
Now,
adding AC both side.
AC + AC = BC + AC
⇒ 2AC = BC + AC
If equals added to equals then whole are equal.
⇒ 2AC = AB [ ∴ AC + BC = AB ]
⇒ AC = ` \frac{1} {2} AB\ `
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.
solution :
Let the given line AB is having two mid points 'C' and 'D'.
Let, AB be the line segment having two mid points 'C' and 'D'.
` \AC=\frac{1} {2} AB\ `____________ (i)
and ` \AD=\frac{1} {2} AB\ `_________ (ii)
Subtracting (i) from (ii),
We have,
` \AC-AD=\frac{1} {2}AB - \frac{1} {2} AB\ `
⇒ ` \AD-AC = 0\ `
⇒ ` \AC=0\ `
∴ C and D is one point.
Thus, every line segment has only one and one mid point.
6. In Fig. 5.10, if AC = BD, then prove that AB = CD.
solution :
Given,
AC = BD
AB + BC = BC + CD
Subtracting BC fron both side,
We get,
AB + BC − BC = BC + CD − BC
AB = CD
Proved.
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.)
solution :
Since this is true for any thing in any part of the world, this is a universal truth.
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