advertisement

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 . 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 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.) 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. 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. 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. 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. 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 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 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 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 Angles opposite to equal sides of an isosceles triangle are equal and inverse is also true Here ∠B = ∠C Sum of All The Interior angles Of a triangle is equal to 1800. ∠A +∠B+ ∠C = 1800 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. 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 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) 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.