Archimedes theory of a circle ABCD and a triangle K

Archimedes theory of a circle ABCD and a triangle K Archimedes compared the area enclosed by a circle to a right triangle whose base has the length of the circle’s circumference and whose height equals the circle’s radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less. He then eliminates each of these by contradiction, leaving equality as the only possibility. Archimedes’ proof consists of constructing a circle ABCD and a triangle K. Archimedes starts by inscribing a square in the circle and bisects the segments of arc AB, BC, CD, DE subtended by the sides of the square. Afterwards he proceeds to inscribe another polygon on the bisected points. He repeats this process until the difference in area between the circle and the inscribed polygon is smaller than the difference between the area of the circle and the area of the triangle. The polygon is then greater than the triangle K. Archimedes then proceeds to explain that a line from the center of the polygon to the bisection of one of its sides is shorter than the radius of the circle, and its circumference is smaller than the circumference of the circle. This disproves the statement that the polygon is greater than the triangle, since the legs of the triangle are made up of the radius and circumference of the circle. The triangle K cannot be both smaller and larger than the polygon, and thus cannot be smaller than the circle. After Archimedes proved that that the triangle cannot be smaller than the circle, he continues to prove that the triangle cannot be larger than the circle, either. This is accomplished by first assuming the triangle K to be larger than the circle ABCD. Then, a square is circumscribed around the circle so that lines drawn from the center of the circle will go through the points A, B, C, and D and bisect the corners of the square, one of which Archimedes labels T. Archimedes then connects the sides of the square with a tangent line and labels the points at which the line meets the square G and F. He goes on to say that because TG GA GH, the triangle formed by FTG is larger than half the area of the difference in area between the square and the circle. Archimedes uses the fact that continual bisecting of the arc of a circle will produce a polygon with this characteristic to assert that continuing this method will ultimately produce a polygon around the circle such that the difference in area between the polygon and the circle is less than the difference in area between the triangle K and the circle. The polygon is thus less in area than the triangle K The length of a line from the center of the circle to a side of the polygon is equal to the radius of the circle. However, the perimeter of the polygon is larger in length than the circumference of the circle, and since the circumference of the circle is equal to the length of the longer leg of the triangle, the polygon must be larger in area than the triangle K. Again, the triangle cannot be both larger and smaller than the polygon, so the triangle cannot be larger than the circle. Archimedes accomplished to prove his theory by using contradiction. After he proved that the triangle with legs equal to the radius and circumference of a given circle is not greater or less in area than that circle he concludes that the two must be equal in area.

Workplace Violence Research Paper Example | Topics and Well Written Essays - 750 words

Workplace Violence - Research Paper Example The growing concern of workplace violence triggered the formation of many labor unions and prevention programs in the country during the 1990s.   Among the prevention programs includes OSHA. Even though labor unions protect employees from workplace violence, many of them are still victims. This is because of lack of effective prevention programs and regulations. For instance, in 1990, OSHA helped in tracking victims of workplace violence. However, it failed to record the number of workplace violence caused by co-workers (Farb & Gordon, 2005). Workplace violence is not only caused by employers but is also caused by co-workers. In the year 1993, NSWI discovered that most of the workplace violence that occurred during that year was caused by economic and psychological problems. Cases of mass killing in workplaces have increased by about 50 percent since 1990. This makes homicide to be the leading cause of workplace psychological and physical injuries in workplaces.   Cases of verbal abuse in the workplaces increased by about 42 between 2005 and 2009. Psychological problems workplaces soon and schizophrenia make people to hurt their co-workers in organizations.   Workplace violence such as verbal abuse and bullying, which do not have negative effects on physical health of employees, has negative effects on their psychological health.   An example of a psychological effect of workplace abuse includes panic attack. 52 percent of employees experience panic attacks in work places.  Ã‚  

Self assignment Term Paper Example | Topics and Well Written Essays - 1500 words - 1

Self assignment - Term Paper Example result, I like to come up with new ways of solving problems, and this often calls for innovation, as such I would like to think of myself as an innovative individual. However, sometimes I tend to go overboard and end up making matter worse with my â€Å"innovation† this is because I can get distracted and in my attempt to use new ways ignore old ones which may be better and more convenient. As I grew up, I was an avid reader of books and I often tried to practice some of the ideas I learnt in real life or to improve on them. For example, in junior high school, I along with 3 of my classmates won the annual science prize for coming up with a homemade solar powered lamp and water heater made from locally available materials including old disused solar panels. This attributes can be explained by my IEI, emotional intelligence test which were rated ENTP, this means I am a creative resourceful and broad minded individual quick on his feet and I can think about a wide range of issues objectively. New ideas exited me and I do not shy away from debating on any issue, since I have no inhibitions about offering my opinion accepting criticism or even giving some. In the IIC5 test, it emerged that I am a patient and strategic individual; I can spend hours on end mulling on a problem until I come up with a solution, as a result, I believe in looking for easier ways of carrying out tasks, but I do not cut corners or ignore protocol in for the sake of quick results. This is because to me it was worth taking time and effort to get good final result than to rush over cutting corners only to come up with substandard answers. This is an approach I have tried and tested over the years more so in my studies where I prefer to strategize patiently and always have a contingency plan in case the original one does not work. Despite this, I sometimes procrastinate and self-deceive that I am being patient while in reality am just avoiding to tackle a problem head on especially when I feel