The Pythagorean Theorem Essay Example | Topics and Well Written Essays - 3500 words

The Pythagorean Theorem - Essay Example The area of such a rectangle is a times b: ab. Therefore the four triangles together are equal to two such rectangles. Their area is 2ab. As for the square whose side is c, its area is simply c. Therefore, the area of the entire square is c + 2ab . . . . . .(1) At the same time, an equal square with side a + b (Fig. 2) is made up of a square whose side is a, a square whose side is b, and two rectangles whose sides are a, b. Therefore the area of that square is a + b + 2ab But this is equal to the square formed by the triangles, line(1): a + b + 2ab = c + 2ab. Therefore, on subtracting the two rectangles -- 2ab -- from each square, we are left with a + b = c. This is the Pythagorean Theorem Proof using similar triangles The Pythagorean theorem, is based on the proportionality of the sides of two similar triangles. Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As so These can be written as Summing these two equalities, we obtain In other words, the Pythagorean theorem: The Arabian mathematician Thabit ibn Kurrah A clever proof by dissection which reassembles two small squares into one larger one was given by the Arabian mathematician Thabit ibn Kurrah (Ogilvy 1994, Frederickson 1997). Proof by Perigal Another proof by dissection is due to Perigal (left...Therefore the four triangles together are equal to two such rectangles. Their area is 2ab. At the same time, an equal square with side a + b (Fig. 2) is made up of a square whose side is a, a square whose side is b, and two rectangles whose sides are a, b. Therefore the area of that square is Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As Another proof by dissection is due to Perigal (left figure; Pergial 1873; Dudeney 1970; Madachy 1979; Steinhaus 1999, pp. 4-5; Ball and Coxeter 1987). A related proof is accomplished using the above figure at right, in which the area of the large square is four times the area of one of the triangles plus the area of the interior square. From the figure d=b-a, so Perhaps the most famous proof of all times is Euclid's geometric proof , although it is neither the simplest nor the most obvious.

Obesity Prevention in America Essay Example | Topics and Well Written Essays - 3000 words

Obesity Prevention in America - Essay Example Realizing the serious public health threat of obesity epidemic prevention focused action plan aimed at decreasing the number of obese children and youth in the United States has been developed by The Institute of Medicine (IOM), which gives explicit goals and recommendations of action plan for preventing obesity and promoting a healthy weight in children and youth in different segments of society based on the best available evidence. As community participation plays a crucial role in promoting health, a collaborative interdisciplinary partnership with community members is expected to further the U.S health initiatives by helping to identify effective strategies that support behavioral change among susceptile groups and implement sustainable program outcomes. Concerted effort by state and federal government, healthcare providers, community participants, as well as individual commitment to make lifestyle modifications will prevent the scourge of obesity and co-morbidities associated wi th it. Obesity is generally associated with consuming more calories than the body needs and imbalance between calories consumed and calories burned, which may be due to genetic, hormonal, behavioral, environmental, or even cultural factors. Overweight and obese people are at increased risk for developing many health risks, such as cardiovascular disease, stroke, hypertension, dyslipidemia, Type-II diabetes, insulin resistance and glucose intolerance, congestive health failure, osteoarthritis, sleep apnea and other respiratory problems, polycystic ovary syndrome, psychological disorders, stress urinary incontinence and kidney stones, cancer of the kidney, endometrium, breast, colon and rectum, etc. (The endocrine society weighs in, 2004, p.6-7). Because of many causes like frequent illness, constant tiredness, and inability to be active due to back pain, joint pain and shortness of breathe the quality and productivity of an obese person is adversely affected, leading to low personal esteem and severe financial crisis. As it is difficult to identify whether a person is obese or not, based on his or her physical appearance, it is essential to differentiate obesity from overweight. An individual with increased body mass index (BMI), a common measure calculation that assesses weight relative to height, which is at least 10 percent over a recommended weight or BMI > 25, is referred to as overweight. Because a patient's abdominal fat is a predictor of risk factors for obesity related medical conditions, physicians consider men and women who have waist measurement greater than 40 inches and 35 inches respectively as overweight. Obesity refers to a body weight that is at least 30 percent over the ideal weight for a specified height, or an individual with a BMI > 30. (The endocrine society weighs in, 2004, p.2). For getting precise information on obesity, in the research setting, physicians use computed tomography, magnetic resonance imaging, or electrical impedance for measuring body fat distribution. Because of the high cost and difficulty in practicing outside laboratory settings these investigations are not adaptable in daily clinical use among healthcare provide rs. However, considering the