Geometry geometry-130805113031-phpapp02

  • Published on
    09-Jun-2015

  • View
    1.287

  • Download
    0

Embed Size (px)

DESCRIPTION

Geometry it talks about the parallel lines and different lines. The target is grade eight to ten. It gives all the basics needed by the learners.

Transcript

<ul><li> 1. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37</li></ul><p> 2. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37 3. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37 4. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37 5. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37 6. Geometry ? Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. It is also a branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties of space.BASICS OF GEOMETRY Point: A point is a location in space. It is represented by a dot. Point is usually named with a upper letter. For example, we refer to the following as "point A Line: A line is a collection of points that extend forever. The following is a line. The two arrows are used to show that it extends forever.We put two points in order to name the line as line AF. However, there are an infinite amount of points. You can also name it line FA.Line segment: A line segment is part of a line. The following is a segment. A segment has two endpoints. The endpoints in the following segments are A and F. Notice also that the line above has no endpoints.SITIKANThA MISHRA Class-VIII Section-A Roll No-37 7. Geometry ? Ray: A ray is a collection of points that begin at one point (an endpoint) and extend forever on one direction. The following is a ray.Angle: Two rays with the same endpoint is an angle. The following is an angle.Plane: A plane is a flat surface like a piece of paper. It extends in all directions. We can use arrows to show that it extends in all directions forever. The following is a plane...Parallel lines When two lines never meet in space or on a plane no matter how long we extend them, we say that they are parallel lines The following lines are parallel..... knjhhjjiIntersecting lines: When lines meet in space or on a plane, we say that they are intersecting lines The following are intersecting lines.Vertex: The point where two rays meet is called a vertex. In the angle above, point A is a vertex.SITIKANThA MISHRA Class-VIII Section-A Roll No-37 8. Geometry ? GEOMOTRICAL FIGURES2D GEOMETRICAL FIGURESA 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane. Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and machine parts made of sheet metal. 2D geometric models are also convenient for describing certain types of artificial images, such as technical diagrams, logos, the glyphs of a font, etc. They are an essential tool of 2D computer graphics and often used as components of 3D geometric models, e.g. to describe the decals to be applied to a car model.SITIKANThA MISHRA Class-VIII Section-A Roll No-37 9. Geometry ?3D GEOMETRYCAL FIGURES3D shapes spheres cubes cones pyramids hemispheres cuboids cylinders These are three-dimensional shapes. Their sides are made of flat or curved surfacesSITIKANThA MISHRA Class-VIII Section-A Roll No-37 10. INDIAN GEOMETREYGeometry ?Vedic periodRigveda manuscript in Devanagari.The Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.[4] The ulba Stras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700-400 BCE) list rules for the construction of sacrificial fire altars.[5] Most mathematical problems considered in the ulba Stras spring from "a single theological requirement,"[6] that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.[6] According to (Hayashi 2005, p. 363), the ulba Stras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." The diagonal rope (akay-rajju) of an oblong (rectangle) produces both which the flank (prvamni) and the horizontal (tiryamn) produce separately."[7] Since the statement is a stra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.[7] They contain lists of Pythagorean triples,[8] which are particular cases of Diophantine equations.[9] They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."[10]SITIKANThA MISHRA Class-VIII Section-A Roll No-37 11. Geometry ? Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3,4,5),(5,12,13),(8,15,17),(7,24,25) and (12,35,37)[11] as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."[11] It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."[11] According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE[12] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[13] indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."[14] Dani goes on to say: "As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in theSulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."[14] In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava (fl. 750-650 BCE) and theApastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.SITIKANThA MISHRA Class-VIII Section-A Roll No-37 12. Classical PeriodGeometry ?In the Bakhshali manuscript, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."[15] Aryabhata's Aryabhatiya (499 CE) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brhma Sphua Siddhnta in 628 CE. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).[16] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:[16] Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).. Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given bywhere s, the semiperimeter, given by: Brahmagupta's Theorem on rational triangles: A triangle with rational sidesfor some rational numbers u,v and w [17]SITIKANThA MISHRA Class-VIII Section-A Roll No-37 13. Geometry ?Father of geometry- EuclidStatue of Euclid in the Oxford University Museum of Natural History.SITIKANThA MISHRA Class-VIII Section-A Roll No-37 14. Geometry ?Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310) Euclid (c. 325-265 BC), of Alexandria, probably a student of one of Platos students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclids was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. 1.Any two points can be joined by a straight line. 2.Any finite straight line can be extended in a straight line. 3.A circle can be drawn with any center and any radius. 4.All right angles are equal to each otherSITIKANThA MISHRA Class-VIII Section-A Roll No-37 15. Geometry ?GEOMETRY IN DAILY LIFEGeometry is used everywhere. Everywhere in the world there is geometry, mostly made by man. Most manmade structures today are in a form of Geometric. How, you ask? Well some examples would the CD, that is a 3-D circle and the case would be a rectangular prism. Buildings, cars, rockets, planes, maps are all great examples. Here's an example on how the world uses Geometry in buildings and structure:-(1)1.This a pictures with some basic geometric structures. This is a modern reconstruction of the English Wigwam. As you can there the door way is a rectangle, and the wooden panels on the side of the house are made up of planes and lines. Except for really planes can go on forever. The panels are also shaped in the shape of squares. The house itself is half a cylinder.2.Here is another modern reconstruction if of a English Wigwam. This house is much similar to the one before. It used a rectangle as a doorway, which is marked with the right angles. The house was made with sticks which was straight lines at one point. With the sticks in place they form squares when they intercepts. This English Wigwam is also half a cylinder.(2)3. This is a modern day skyscraper at MIT. The openings and windows are all made up of parallelograms. Much of them are rectangles and squares. This is a parallelogram kind of building.(3) SITIKANThA MISHRA Class-VIII Section-A Roll No-37 16. Geometry ? 4. This is the Hancock Tower, in Chicago. With this image, we can show you more 3D shapes. As you can see the tower is formed by a large cube. The windows are parallelogram. The other structure is made up of a cone. There is a point at the top where all the sides meet, and There is a base for it also which makes it a cone. 5. This is another building at MIT. this building is made up of cubes, squares and a sphere. The cube is the main building and the squares are the windows. The doorways are rectangle, like always. On this building There is a structure on the room that is made up of a sphere. 6.This is the Pyramids, in Indianapolis. The pyramids are made up of pyramids, of course, and squares. There are also many 3D geometric shapes in these pyramids. The building itself is made up of a pyramid, the windows a made up of tinted squares, and the borders of the outside walls and windows are made up of 3D geometric shapes. 7. This is a Chevrolet SSR Roadster Pickup. This car is built with geometry. The wheels and lights are circles, the doors are rectangular prisms, the main area for a person to drive and sit in it a half a sphere with the sides chopped off which makes it 1/4 of a sphere. If a person would look very closely the person would see a lot more shapes in the car. Too many to list.SITIKANThA MISHRA Class-VIII Section-A Roll No-37 17. Geometry ?SymmetrySymmetry in common usage generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise. Leonardo da Vinci's Vitruvian Man (1492) is often used as a representation of symmetry in the human body and, by extension, the natural universe. Reflection symmetry, Rotational symmetry is symmetry with respect to some or all rotations in mdimensional Euclidean space Helical symmetry is the kind of symmetry seen in such everyday objects as springs, Slinky toys, drill bits, and augers. It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis).Symmetry in religious symbols Symmetry in architecture (Eg. Qutub Minar, etc) A drill bit with helical symmetrySITIKANThA MISHRA Class-VIII Section-A Roll No-37 18. Geometry ? IMPORTANCE OF GEOMETRY Geometry must be looked at as the consummate, complete and paradigmatic reality given to us inconsequential from the Divine Revelation. These are the reasons why geometry is important: It hones one's thinking ability by using logical reasoning. It helps develop skills in deductive thinking which is applied in all other fields of learning. Artists use their knowledge of geometry in creating their master pieces. It is a useful groundwork for learning other branches of Mathematics. Students with knowledge of Geometry will have sufficient skills abstracting from the external world. Geometry facilitates the s...</p>