19. May 2018•0 gefällt mir•2,879 views

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Ibrahim KhanFolgen

- 1. By Ms.Arshi
- 2. Introduction Geometry:-’Geo’ means “earth” and ‘metrein’ means “to measure”. Need of measuring land
- 3. Evolution of Geometry in different countries(Egypt) River Nile flooding- boundaries of plots were overdrawn Techniques for finding volumes of granaries, constructing canals and pyramids.
- 4. Truncated Pyramid -Volume
- 5. Use of Geometry in Indus Valley Civilization Excavations at Harappa and Mohenjodaro. Cities , roads, drainage systems, rooms of different types(mensuration and practical arithmetic), bricks (4:2:1)
- 6. Geometry was in
- 7. Thales (Greek Mathematician) Thales gave first known proof Circle is bisected by its diameter
- 8. Pythagoras Theorem Pythagoras
- 9. Euclid (325-265 BC) “Elements” Treatise on Math & Geometry
- 10. Euclid’s Definitions 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.
- 11. The definitions of a point, a line, and a plane, are not accepted by mathematicians. Therefore, these terms are taken as undefined. Models Euclid’s Definitions problem
- 12. Assumptions specific to Geometry Common notions not linked to Geometry
- 13. Euclid's Postulates: 1. A straight line can be drawn from any point to any point. 2. A terminated line can be produced indefinitely. 3. It is possible to describe a circle with any centre any distance. 4. All right angles are equal to one another. 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.
- 14. Two equivalent versions of the Fifth Euclid’s postulates: (i). ‘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’. (ii). Two distinct intersecting lines cannot be parallel to the same line.
- 15. Euclid's Axioms: 1. Things which are equal to the same things are also equal to one another. E.g. If A=B & C=B. that is both A & C are equal to B, then A & C will be equal. 2. If equals are added to equals, then the wholes are equal. E.g. Two glasses A & A’ has same volume of water. Now we add equal quantity of water B, to both glass A & A’, then the final volume of water in the jar will be same. A+ B will be equal to A’ + B. 3. If equals are subtracted from equals, then the remainders are equal. E.g. Two glasses A & A’ has same volume of water. Now we remove equal quantity of water B, from both glass A & A’, then the final volume of water in the jar will be same. A- B will be equal to A’ - B. 4. Things which coincide with one another are equal to one another. E.g. if two triangles coincide with each other then they are equal.
- 16. Euclid's Axioms: 5. The whole is greater than the part. This statement is true in physics, chemistry, mathematics, geometry, biology, economics etc. 6. Things which are double of the same things are equal to one another. If A= A’ then 2A= 2A’. 7. Things which are halves of the same things are equal to one another. If A= A’ then ½ A= ½ A’.
- 17. Theorems or Prepositions: After stating Postulates & Axioms, Euclid used these to prove other results by applying deductive reasoning. E.g.: “Diameter divides circle in 2 parts” is a theorem. Euclid deduced 465 Theorems. “Two distinct lines cannot have more than one point in common” is a theorem.
- 18. Non Euclidean Geometry/Spherical geometry Lines are not straight. They are parts of great circles (i.e., circles obtained by the intersection of a sphere and planes passing through the centre of the sphere). 5th postulate of Euclid- Lines AN & BN should not meet but they meet at point N. It has been proved that Euclidean geometry is valid only for the figures in the plane. On the curved surfaces, it fails.