The
word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’, and
‘metrein’, meaning ‘to measure’. Geometry appears to have originated from the
need for measuring land.
the
Egyptians developed a number of geometric techniques and rules for calculating
simple areas and also for doing simple constructions.
The
knowledge of geometry was also used by them for computing volumes of granaries,
and for constructing canals and pyramids. They also knew the correct formula to
find the volume of a truncated pyramid
In the Indian subcontinent, the excavations at Harappa
and Mohenjo-Daro . The houses had many rooms of different types. This shows
that the town dwellers were skilled in mensuration and practical arithmetic.
The bricks used for constructions were kiln fired and the ratio length :
breadth : thickness, of the bricks was found to be 4 : 2 : 1.
The sriyantra (given
in the Atharvaveda) consists of nine
interwoven isosceles triangles. These triangles are arranged in such a way that
they produce 43 subsidiary triangles.
Euclid summarised
these statements as definitions. He began his exposition by listing 23
definitions in Book 1 of the ‘Elements’. A few of them are given below :
1. A
point is that which has no part.
2. A
line is breadthless length.
3. The
ends of a line are points.
4. A
straight line is a line which lies evenly with the points on itself.
5. A
surface is that which has length and breadth only.
6. The
edges of a surface are lines.
7. A
plane surface is a surface which lies evenly with the straight lines on itself.
Starting
with his definitions, Euclid assumed certain properties, which were not to be
proved. These assumptions are actually ‘obvious universal truths’. He divided
them into two types: axioms
and postulates.
Some of Euclid’s axioms, not in his order, are given below :
(1) Things which are equal to
the same thing are equal to one another.
(2) If equals are added to
equals, the wholes are equal.
(3) If equals are subtracted
from equals, the remainders are equal.
(4) Things which coincide with
one another are equal to one another.
(5) The whole is greater than
the part.
(6) Things which are double of
the same things are equal to one another.
(7) Things which are halves of
the same things are equal to one another.
Now let us discuss Euclid’s five postulates. They are :
Postulate 1 : A straight line may be drawn
from any one point to any other point.
Postulate
2 : A terminated line can be produced indefinitely.
Postulate
3 : A circle can be drawn with any centre and any radius.
Postulate
4 : All right angles are equal
to one another.
Postulate 5 : If a straight line falling
on two straight lines makes the interior angles on the same side of it taken together less than two right
angles, then the two straight lines, if produced indefinitely, meet on that
side on which the sum of angles is less than two right angles.
Example 1 : If A, B and C are three points on
a line, and B lies between A and C (see Fig. 5.7), then prove that AB + BC =
AC.
Fig. 5.7
Solution : In the figure given
above, AC coincides with AB + BC.
Also, Euclid’s Axiom (4) says that things which coincide
with one another are equal to one another. So, it can be deduced that
AB + BC = AC
Note that in this solution, it has been assumed that there
is a unique line passing through two points.
Example 2 : Prove that an
equilateral triangle can be constructed on any given line segment.
Solution : In the statement above, a line
segment of any length is given, say AB [see Fig. 5.8(i)].
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Fig. 5.8
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Here, you need to do some
construction. Using Euclid’s Postulate 3, you can draw a circle with point A as
the centre and AB as the radius [see Fig. 5.8(ii)]. Similarly, draw another
circle with point B as the centre and BA as the radius. The two circles meet at
a point, say C. Now, draw the line segments AC and BC to form ∆ ABC
[see Fig. 5.8 (iii)].
So, you have to prove that this triangle is equilateral,
i.e., AB = AC = BC.
Now, AB = AC, since they are the radii
of the same circle (1)
Similarly, AB = BC (Radii
of the same circle) (2)
From these two facts, and
Euclid’s axiom that things which are equal to the same thing are equal to one
another, you can conclude that AB = BC = AC.
So, ∆
ABC is an equilateral triangle.
Note that here Euclid has
assumed, without mentioning anywhere, that the two circles drawn with centres A
and B will meet each other at a point.
Now we prove a theorem, which is
frequently used in different results:
Theorem 5.1 : Two distinct lines
cannot have more than one point in common.
Proof : Here we are given two
lines l and m. We need to prove that they have only one point in common.
For the time
being, let us suppose that the two lines intersect in two distinct points, say
P and Q. So, you have two lines passing through two distinct points P and Q.
But this assumption clashes with the axiom that only one line can pass through
two distinct points. So, the assumption that we started with, that two lines
can pass through two distinct points is wrong.
From this, what can we conclude? We are
forced to conclude that two distinct lines cannot have more than one point in
common.
Axiom 5.1 : Given two distinct
points, there is a unique line that passes through them.
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Fig. 5.4
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EXERCISE 5.1
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. FALSE
(ii) There are
an infinite number of lines which pass through two distinct points. FALSE
(iii) A
terminated line can be produced indefinitely on both the sides. TRUE
(iv) If two
circles are equal, then their radii are equal. TRUE
(v) In Fig.
5.9, if AB = PQ and PQ = XY, then AB = XY.
TRUE
(1)
Things which are equal to the same
thing are equal to one another.
Fig. 5.9
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=NEVER INTRESECT EACHOTHER
(ii) perpendicular lines=TWO LINES
INTESRSECT EACHOHER AT 900
(iii) line segment=A
STRAIGHT LINE DRAWN FROM ANY POINT TO ANY OTHER POINT IS CALLED AS LINE
SEGMENT.
(iv) radius of a circle=IT IS THE DISTANCE BETWEEN
THE CENTER OF A CIRCLE TO ANY POINT LYING ON THE CIRCLE.
(v) square=A SQUARE
IS A QUADRILATERAL HAVING ALL SIDES OF EQUAL LENGTH AND ALL ANGLES OF SAME
MEASURE i.e 900
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.
Various undefined terms . they
don’t follow from Euclid’s postulates they follow from 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 =
AB. Explain by drawing
the figure.
Things which
are equal to the same thing are equal to one another.
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.
6. In Fig.
5.10, if AC = BD, then prove that AB = CD.
Fig. 5.10
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.)
‘For every line l and for every point P not lying on
l, there exists a unique line m passing through P and parallel to l’.
From Fig.
5.11, you can see that of all the lines passing through the point P, only line
m is parallel to line l.
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Fig. 5.11
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This
result can also be stated in the following form:
Two distinct intersecting lines cannot be parallel
to the same line.
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