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Geometry geometry


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Geometry geometry

  1. 1. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  2. 2. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  3. 3. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  4. 4. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  5. 5. Mathematics Project (Geometry) SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  6. 6. 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. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  7. 7. 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.……..……………………………………….. knjhhjji Intersecting 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. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  8. 8. GEOMOTRICAL FIGURES- 2D GEOMETRICAL FIGURES- A 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. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  9. 9. 3D GEOMETRYCAL FIGURES- These are three-dimensional shapes. Their sides are made of flat or curved surfaces 3D shapes spheres cubes cones pyramids hemispheres cuboids cylinders Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  10. 10. INDIAN GEOMETREY Vedic period Rigveda manuscript in Devanagari. The Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.[4] The Śulba Sūtras (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 Śulba Sūtras 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 Śulba Sūtras 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 (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."[7] Since the statement is a sūtra, 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] Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  11. 11. 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. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  12. 12. 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 Brāhma Sphuṭa Siddhānta 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).…………………………………………………………………. Classical Period Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by where s, the semiperimeter, given by: Brahmagupta's Theorem on rational triangles: A triangle with rational sides for some rational numbers u, v and w [17] Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  13. 13. Father of geometry-Euclid Statue of Euclid in the Oxford University Museum of Natural History. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  14. 14. 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 Plato’s 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 Euclid’s 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 other Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  15. 15. Geometry 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. GEOMETRY IN DAILY LIFE Here's an example on how the world uses Geometry in buildings and structure:- 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. (1) 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) Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  16. 16. 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. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  17. 17. Symmetry Symmetry 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 m- dimensional 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 symmetry Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  18. 18. 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 solution of problems from other fields since its principles are applicable to other disciplines. •Knowledge of geometry is the best doorway towards other branches of Mathematics. •It can be used in a wide array of scientific and technical field. The importance of Geometry is further substantiated by the requirement that it is incorporated as a basic subject for all college students. An educated man has within his grasps mathematical skills together with the other qualities that make him a gentleman. Finally, what is the importance of Geometry? From a philosophical point of view, Geometry exposes the ultimate essence of the physical world. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  20. 20. How Is Geometry Used in Real Life? Understanding of geometry takes years of study and involves many related sub-fields. Geometry is the mathematics of space and shape, which is the basis of all things that exist. Understanding geometry is a necessary step in understanding how the world is built. Most people take geometry in high school and learn about triangles and vertical angles. The application of geometry in real life is not always evident to teenagers, but the reality is geometry infiltrates every facet of our daily living. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  21. 21. Geometry and Children Geometry is not generally covered in grades kindergarten through eight, but children are introduced to shapes and spaces in a variety of ways. In initial school activities, kindergarten students are asked to color triangles and circles. By the end of elementary school, most students are able to make scale drawings. Students are able to connect locations with coordinates, which is analytical geometry. Visualization and spatial reasoning skills assist students with problem solving. Geometry in the Real World In the real world, geometry is everywhere. A few examples include buildings, planes, cars and maps. Homes are made of basic geometric structures. Some skyscrapers have windows made of rectangles and squares. The John Hancock Tower in Chicago is made of a long cube. On a car, the wheels and lights are circles. The great pyramids of Egypt of made of geometric shapes. Symmetry in Science Symmetry is a sense of harmony, proportion and balance. It reflects beauty and perfection. In the scientific sense, symmetry is defined as a sense of self-similarity through rules of a formal system, such as geometry or physics. Symmetry is the basic concept in the study of biology, chemistry and physics. Systems of laws in physics and molecules in stereo chemistry reflect the concepts of geometry. Some have difficulty grasping how geometry relates to sciences. Since the 1870s, the study of transformation and related symmetry are parallel to geometric studies. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  22. 22. Along with new manufacturing, several changes have been made including a re-designed upper control arm and other updates to weight, strength and function. “Increased demand for pre-engineered IRS suspensions has prompted us to update and continue our line of Advanced Geometry Systems for the 1999-2004 SVT Cobra", according to Kenny Brown. “The quantum-leap forward in technology that the Cobra IRS represents provides the perfect platform for open-track enthusiasts who can now find these cars for a reasonable price. We raced IRS Cobras starting in 1999 all the way through 2004, and we still have several customers who compete with it today. We know exactly how to make that architecture work for the street and race track, and we are the only company that has ever supported it whole heartedly". Kenny’s Advanced Geometry Independent Rear Suspension is designed to replace the OEM suspension components offering; racing-inspired suspension geometry, 40 percent reduction in weight, and conversion to coil-over shock design. A complete independent rear suspension upgrade consists of; Tubular Rear Lower Control Arms, Tubular Rear Upper Control Arms, IRS Rear Steer Kit, IRS Forward Torque Brace, Aluminum IRS Differential Bushings, and Coil-Over Rear Upper Shock Mounts. There are also a range of coil-over shocks available depending on the application. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  23. 23. The extended performance benefits of the Kenny Brown Advanced Geometry Independent Rear Suspension are; improved handling, reduced wheel-hop, eliminates rear steer, reduced weight, improved strength, and clearance for coil-over shocks. Everything you need to transform the car for aggressive street or full blown open-track is available in one simple system, or combination of components depending upon your application. Tubular Rear Upper and Lower Control Arms are designed to replace the 1999-2004 Cobra OEM control arms and are available for street and competition applications. The tubular rear control arms feature urethane bushings for reduced deflection and upgraded rear sway bar links for improved strength and performance. The new control arms eliminate the rear spring seats, allowing for conversion to coil-over shocks and much greater adjustability. IRS Rear Steer Kit is designed to replace the OEM rear tie rods and inner tie-rod ends with heavy-duty competition-grade hardware. The rear steer kit eliminates the rear steer factor allowing the car to exit corners better with improved grip and helps to eliminate wheel-hop. IRS Forward Torque Brace is designed to strengthen the area where the differential assembly mounts to the IRS carrier assembly. The forward torque brace improves traction and helps eliminate wheel-hop. Aluminum IRS Differential Bushing Kit is designed to replace the OEM rubber bushings to eliminate deflection at the rear differential housing. The aluminum bushings improve traction, help eliminate wheel-hop and act as a heat soak to pull critical temperature away from the fragile rear differential. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  24. 24. Coil-Over Rear Upper Shock Mounts attach at the OEM rear upper shock mount and allow for fitment of competition coil-over rear shocks. The rear shock mounts are bolt-on and will work for most aftermarket coil-over shocks. When running coil-over shocks in the rear it is strongly recommended that you also use the Heavy-Duty Rear Shock Tower Brace. Advanced Geometry Independent Rear Suspension Components for 1999-2004 SVT Cobra Mustangs along with other world-class chassis and suspension components for the popular SN-95 platform are available through authorized Kenny Brown Performance Parts Dealers, online at or by calling Kenny Brown Performance direct – (855) 847-4477. Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
  25. 25. GEOMETRY AND EARTH An interesting topic in 3-dimensional geometry is Earth geometry. The Earth is very close to a sphere (ball) shape, with an average radius of 6371 km. (It's actually a bit flat at the poles, but only by a small amount). Earth geometry is a special case of spherical geometry. When we measure distances that a boat or aircraft travels between any 2 places on the Earth, we do not use straight line distances, since we need to go around the curve of the Earth from one place to another. (Think about the direct or straight-line distance between London and Sydney, through the Earth. That's going to be a lot less than the distance a plane flies around the surface of the Earth.) Let's start with an example. What distance does a plane fly between Beijing, China and Perth, Western Australia? Geometry ? SITIKANThA MISHRA Class-VIII Section-A Roll No-37
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