A project to study the
Geometry- A Branch of Mathematics
Prepared By:
Ankush Kumar
Class: IX C
School: K V No 1 Bti Cantt.
Mathematics is a branch of science that deals
with the detailed study of the numbers and
calculations , in order to solve many daily life
problems .
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Geometry (Ancient
Greek: γεωμετρία; geo"earth", metri "measurement")
"Earth-measuring" is a
part of
mathematics concerned
with questions of size,
shape, relative position of
figures, and the properties
of space. Geometry is one
of the oldest sciences.
Oxyrhynchus papyrus (P.Oxy. I 29)
showing fragment of Euclid's Elements
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Geometry (Greek γεωμετρία;
geo = earth, metria = measure)
arose as the field of knowledge
dealing with spatial relationships.
Geometry was one of the two fields
of pre-modern mathematics, the
other being the study of numbers.
An ancient geometric chart
Classic geometry was focused in
compass and straightedge
constructions. Geometry was revolutionized
by Euclid, who introduced mathematical rigor and the axiomatic
method still in use today. His book, The Elements is widely considered
the most influential textbook of all time, and was known to all
educated people in the West until the middle of the 20th century.
In modern times, geometric concepts have been generalized to a high
level of abstraction and complexity, and have been subjected to the
methods of calculus and abstract algebra, so that many modern
branches of the field are barely recognizable as the descendants of
early geometry. (See areas of mathematics and algebraic geometry.)
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Geometry originated as
a practical science concerned
with surveying, measurements,
areas, and volumes. Among the
notable accomplishments one
finds formulas
for lengths, areas and volumes,
such asPythagorean
theorem, circumference and area
of a circle, area of a triangle,
volume of a cylinder, sphere, and
a pyramid. A method of
computing certain inaccessible
distances or heights based
on similarity of geometric figures
is attributed to Thales.
Development of astronomy led to
emergence of trigonometry and
spherical trigonometry, together
with the attendant computational
techniques.
Visual proof of the Pythagorean
theorem for the (3, 4, 5)triangle as in
the Chou Pei Suan Ching 500–200 BC.
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The theme of symmetry in geometry is nearly as old
as the science of geometry itself. The circle, regular
polygons and platonic solids held deep significance
for many ancient philosophers and were investigated
in detail by the time of Euclid. Symmetric patterns
occur in nature and were artistically rendered in a
multitude of forms, including the bewildering
graphics of M. C. Escher. Nonetheless, it was not
until the second half of 19th century that the unifying
role of symmetry in foundations of geometry had
been recognized. Felix Klein's Erlangen
program proclaimed that, in a very precise sense,
symmetry, expressed via the notion of a
transformation group, determines what geometry is.
Symmetry in classical Euclidean geometry is
represented by congruences and rigid motions,
whereas in projective geometry an analogous role is
played by collineations, geometric transformations
that take straight lines into straight lines.
A tiling of the hyperbolic
plane
A triangle is one of the
basic shapes of geometr
y: a polygon with three
corners or vertices and
three sides or edges
which are line
segments. A triangle
with vertices A, B,
and C is
denoted ∆ ABC.
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Types of Triangles:
1) On The Basis Of Lengths Of Sides
2) On The Basis Of Internal Angles
Basic Facts
Congruence Of Triangles
Properties Of Triangles
Pythagoras Theorem
Herons Formula
Summary
In a scalene triangle,
all sides are
unequal. The three
angles are also all
different in measure.
Notice that a scalene
triangle can be (but
need not be) a right
triangle.
In an isosceles triangle, two
sides are equal in length. An
isosceles triangle also has
two angles of the same
measure; namely, the angles
opposite to the two sides of
the same length; this fact is
the content of the Isosceles
triangle theorem. Some
mathematicians define an
isosceles triangle to have
exactly two equal sides,
whereas others define an
isosceles triangle as one
with at least two equal
sides. The latter definition
would make all equilateral
triangles isosceles triangles.
In an equilateral
triangle all sides have the
same length. An
equilateral triangle is also
a regular polygon with all
angles measuring 60°
A triangle that has all
interior angles
measuring less than
90° is an acute
triangle or acuteangled triangle.
A right triangle (or right-angled
triangle, formerly called a rectangled
triangle) has one of its interior angles
measuring 90° (a right angle). The side
opposite to the right angle is
the hypotenuse; it is the longest side
of the right triangle. The other two
sides are called the legs or catheti.
(singular: cathetus) of the triangle.
Right triangles obey the Pythagorean
theorem: the sum of the squares of
the lengths of the two legs is equal to
the square of the length of the
hypotenuse: a2 + b2 = c2,
where a and b are the lengths of the
legs and c is the length of the
hypotenuse. Special right triangles are
right triangles with additional
properties that make calculations
involving them easier. The most
famous is the 3-4-5 right triangle,
where 32 + 42 = 52. In this situation,
3, 4, and 5 are a Pythagorean Triple.
A triangle that has one
angle that measures more
than 90° is an obtuse
triangle or obtuse-angled
triangle.
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Triangles are assumed to be two-dimensional plane figures,
unless the context provides otherwise (see Non-planar triangles,
below).
The sum of the lengths of any two sides of a triangle always
exceeds the length of the third side, a principle known as
the triangle inequality. Since the vertices of a triangle are
assumed to be non-collinear, it is not possible for the sum of the
length of two sides be equal to the length of the third side.
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SAS Postulate:
Two sides in a
triangle have the
same length as
two sides in the
other triangle, and
the included
angles have the
same measure.
Here,
AB = PQ
∠BAC = ∠PQR
AC = QR
So, ∆BAC ≌ ∆PQR,
by SAS Congruence Rule
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ASA: Two interior
angles and the
included side in a
triangle have the
same measure and
length ,
respectively, as
those in the other
triangle.
(The included side
for a pair of angles
is the side that is
common to them.)
Here,
∠ABC = ∠QPR
AB = PQ
∠BAC =∠ PQR
So, ∆CAB ≌ ∆RQP
By, ASA
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SSS: Each side
of a triangle
has the same
length as a
corresponding
side of the
other triangle.
Here,
AB = PQ
BC = QR
AC=PR
So, ∆ABS ≌ ∆PQR
By SSS Congruence
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RHS: If in two
triangles the
hypotenuse and
one side of one
triangle are equal
to the hypotenuse
and one side of
other triangle then
the two triangles
are congruent.
Here,
∠ABC = ∠PQR
AC = PR
BC = QR
So, ∆ABC ≌ ∆PQR,
By RHS Congruence
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Angles opposite to equal sides of an isosceles
triangle are equal and inverse is also true
Here ∠B = ∠C
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Sum of All The Interior angles Of a triangle is
equal to 1800.
∠A +∠B+ ∠C = 1800
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Exterior angle of a triangle is equal to the
sum of interior opposite angles.
∠BAC +∠ABC = ∠ACD
Since the fourth century AD, Pythagoras has commonly
been given credit for discovering the
Pythagorean theorem, a theorem in geometry that states
that in a right-angled triangle the square of the hypotenuse
(the side opposite the right angle), c, is equal to the sum
of the squares of the other two sides, b and a—that is
, a 2 + b 2 = c 2.
While the theorem that now bears his name was known
and previously utilized by the Babylonians and Indians, he, or his students, are
often said to have constructed the first proof. It must, however, be stressed
that the way in which the Babylonians handled Pythagorean numbers, implies
that they knew that the principle was generally applicable, and knew some kind
of proof, which has not yet been found in the (still largely
unpublished) cuneiform sources. Because of the secretive nature of his school
and the custom of its students to attribute everything to their teacher, there is
no evidence that Pythagoras himself worked on or proved this theorem. For
that matter, there is no evidence that he worked on any mathematical or metamathematical problems. Some attribute it as a carefully constructed myth by
followers of Plato over two centuries after the death of Pythagoras, mainly to
bolster the case for Platonic meta-physics, which resonate well with the ideas
they attributed to Pythagoras. This attribution has stuck, down the centuries up
to modern times. The earliest known mention of Pythagoras's name in
connection with the theorem occurred five centuries after his death, in the
writings of Cicero and Plutarch.
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In a right triangle the
square of the
hypotenuse of the
triangle is equal to the
sum of the squares of
the remaining sides in
that triangle.
AC2 = AB2 + BC2 ,
i.e. H2 = P2 + B2
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Heron was born in about 10AD
possibly in Alexandria in Egypt. He
worked in applied mathematics. His
works on mathematics and physical
subjects are so numerous and
varied that he is considered to be
an encyclopedic writer in these
fields. His geometrical works deal
largely
with
problems
on
mensuration written in three books.
Book I deals with the area of
squares,
rectangles,
triangle,
trapezoids( trapezia ) various other
specialized
quadrilaterals,
the
regular polygon, circles, surface of
cylinder, cones, spheres etc. in this
book, heron has derived the famous
formula for the area of triangle in
terms of its three sides.
Heron(10AD - 75AD)
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To find the area of a triangle with the help of
given sides and no perpendicular bisector ,
Heron framed a formula called, Heron’s Formula.
Area of triangle =
Where , a , b and c are the sides of a triangle
And s=(a+b+c)/2
Area of Triangle = ½(base*height)
=
 Angle sum Prop,
∠A +∠B+ ∠C = 1800
 Exterior Angle Prop,
ext angle = sum of interior opp angles
 Four rules to prove congruence are
SAS ,SSS , ASA , RHS.
Also there is one more rule called AAS i.e.
Angle Angle Side Congruence Rule.
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