Lesson no. 5 . Introduction to Euclid's Geometry
Geo- Earth
Metry- Measure
Geometry appear to have originated from the need for measuring land.
Euclid's Geometry-
It is the study of plane and solid figure on the basis of axioms, theorems , postulates compelled by Euclid in his book "The element"
Untilled 1900 geometry implied euclidean geometry as there was no other form of geometry at that time. Recently non- euclidean geometry system has been developed.
Euclid's Definitions-
Euclid give more than hundred definition in the book
I) A point is that which has no part.
ii) A line is breathless length.
iii) A surface is that have length and breadth only .
iv) parallel line are those which do not meet.
ii) A line is breathless length.
iii) A surface is that have length and breadth only .
iv) parallel line are those which do not meet.
Axioms-
Axioms are common notion, any discipline not only restricted to geometry .These are universal truth some Axioms are given below:
I) whole is greater than a part.
ii)things which are equal to same thing are equal to each other.If a=x,b =x than ,a=b
iii) if equal are added to equal then the whole are equal. If a = b than, a +x= b+x
Postulates:
Postulates are specific to Euclid geometry. There are five postulates given below:
I) The unique straight line may be drawn from any point to any other point.
ii) A terminated line can be produced in definitely .
iii) A circle can be drawn with any centre and any radius.
iv) All right angles are equal to one another.
v) If a straight line falling onto a straight line makes the interior angle , on the some side of it taken together less than two right angle, then the two straight lines produced in definitely meet on that side in which sum of angle less than two right angles.
Consistent:
It was not possible to diduce any statement using this Axioms, that contradict any axioms ,This this system is taken as consistent.
Propositious/ Theroms :
Statement that were true with the help of axioms and postulate are called theorem.
Ex . Diameter divided circle into parts.
Exercise 5.1
Q1. Give 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
Ans : parallel lines are straight lines which are in the same plane and perpendicular distance between this line is always constant and never intersect each other.
Ans : parallel lines are straight lines which are in the same plane and perpendicular distance between this line is always constant and never intersect each other.
ii) perpendicular line
Ans: Two lines are said to be perpendicular line if the angle between them is a right angle.
Ans: Two lines are said to be perpendicular line if the angle between them is a right angle.
iii) line segment
Ans : a straight line drawn from any point to any other point is called a line segment.
iv) radius of a circle
Ans:it is the length of a line segment joining the centre of circle to any point on the circle.
v) square
Ans: a square is a 4 sided figure which is both equilateral and right angle.
Q2. 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.
Given,
AC =BC
To prove that AC= 1\2 AB
Proof : it is given that AC= BC ...........{1 }
Since , C lies between A& B
AC +AC =AB
2AC= AB
AC =1/2 AB. [ From eq { 1}]
Q 3. In question 4 point C is called a midpoint of line segment AB .prove that every line segment has one and only one midpoint.
Ans:
Let C&D are two midpoint of AB .
C is the midpoint of AB
AC =BC .............{1}
C lies between A & B ...... [from eq 1]
AC +CB= AB
AC+ AC =AB
2AC =AB ................{2 }
Since, D is the midpoint of AB
AD= BD
D lies between A&B
AD +DB =AB
AD +AD =AB
2AD =AB .............{3}
[From eq 2 & 3]
2AC = 2AD
AC=AD .
[Things which are halve of the same things are equal to one another.]
Ans: a square is a 4 sided figure which is both equilateral and right angle.
Q2. 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 that AC= 1\2 AB
Proof : it is given that AC= BC ...........{1 }
Since , C lies between A& B
AC +AC =AB
2AC= AB
AC =1/2 AB. [ From eq { 1}]
Q 3. In question 4 point C is called a midpoint of line segment AB .prove that every line segment has one and only one midpoint.
Ans:
Let C&D are two midpoint of AB .
C is the midpoint of AB
AC =BC .............{1}
C lies between A & B ...... [from eq 1]
AC +CB= AB
AC+ AC =AB
2AC =AB ................{2 }
Since, D is the midpoint of AB
AD= BD
D lies between A&B
AD +DB =AB
AD +AD =AB
2AD =AB .............{3}
[From eq 2 & 3]
2AC = 2AD
AC=AD .
[Things which are halve of the same things are equal to one another.]
