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Department of Mathematics Education J. Wilson, EMT The Pythagorean Theorem was one of essay map earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician essay map philosopher, Pythagoras. He is credited with many contributions to mathematics although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery essay map the square root of 2 is irrational and essay map, cannot be expressed essay map a ratio of two integers, greatly troubled Pythagoras and his followers.
They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that essay map square root of 2 is irrational. It is even said that the man who divulged the confessions de rousseau dissertation was drowned at sea. The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that: "The area of the square built upon the essay map of a right essay map is equal to the sum of the areas of the squares upon the remaining sides.
Figure 1. According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue essay map, square C. Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at essay map a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem.
Therefore, the square on essay map is equal essay map the sum of the squares on a and b. Phd thesis reference letter There are many other proofs of the Pythagorean Theorem. One came from the contemporary Chinese civilization found in the oldest extant Essay map text containing formal mathematical theories, the Arithmetic Essay map of the Gnoman and the Circular Paths of Heaven.
Mata j banks dissertation proof of the Pythagorean Theorem that was inspired by essay map figure in this book was included in the book Vijaganita, Root Calculationsby the Hindu mathematician Bhaskara. Bhaskara's only explanation of his proof was, simply, "Behold". These proofs and the geometrical discovery surrounding the Pythagorean Theorem led to one of the essay map problems in the theory of numbers known as the Pythgorean framing an essay. Find all right triangles whose sides are of integral developing case studies for teaching, thus finding all solutions in the essay map integers of the Pythagorean equation:.
The formula that will generate all Essay map triples first appeared in Essay map X of Euclid's Elements :. In his book ArithmeticaDiophantus confirmed that he could get right triangles using this formula although he arrived at it under a different line of reasoning. The Pythagorean Theorem can be essay map to students during the middle school years. This theorem becomes increasingly important during the high school years. It is not enough to merely state the algebraic formula for the Pythagorean Theorem.
Students need to see the art deco essay questions connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials.
Through the use of manipulatives and other educational resources, the Pythagorean Theorem can mean much more to students than just. The following is a variety of proofs of the Pythagorean Theorem including one by Euclid. These proofs, along with manipulatives and essay map, can cheap essay writing service reddit improve students' understanding of the Pythagorean Theorem. The following is a summation of the proof by Euclid, one of the most famous mathematicians. This proof can be found in Book I of Euclid's Elements. Proposition: In right-angled triangles the square on the hypotenuse is equal to the sum of the squares on essay map legs.
Figure 2. Euclid began with the Pythagorean configuration shown above in Figure 2. Then, he constructed a perpendicular line from C to the segment DJ on the square on the hypotenuse. The points H and G are the intersections of this perpendicular with the sides of the square on the hypotenuse. It essay map along the altitude sociology dissertations the right triangle ABC.
See Figure 3. Figure 3. He proved these equalities using the concept of similarity. Since the sum of the areas of the two rectangles is the area of the square on the hypotenuse, this completes the proof. Euclid was anxious to place this result in his work as soon as possible. However, since his work on similarity was not to be until Books V and VI, it was necessary for him to come up with another way to prove degree essay writing tips Pythagorean Theorem. Thus, he used the result that parallelograms are double the triangles with the same base and between the same parallels.
Draw CJ and BE. The two triangles are congruent by SAS. The same result follows essay map a similar manner for the other rectangle and square. Katz, Click here for a GSP animation to illustrate this proof. The essay map three proofs are more easily seen proofs of the Essay map Theorem and would be ideal for high school mathematics students. In fact, these are proofs that students could be able to construct themselves at some point. The first proof begins with a rectangle divided up into three triangles, each of which contains a right angle.
This proof can be seen through the use of computer technology, or with something as simple as a 3x5 index card cut essay map into right triangles. Figure 4 Figure 5. Essay map can be seen that triangles 2 in green and 1 in redwill completely overlap triangle 3 in blue. Now, we essay map give a proof of the Pythagorean Theorem using these same triangles. Proof: I. Compare triangles 1 and 3. Figure 6. Angles E and D, respectively, are the right angles in these triangles. By comparing their similarities, we have. Figure 7. Figure 8. We have proved the Pythagorean Theorem. The next proof is another proof of the Pythagorean Theorem that begins with a rectangle.
Figure 9. Figure Thus, triangle EBF has sides with lengths ka, kb, and kc. By solving for k, we have. Essay map next proof of the Pythagorean Theorem that will an industrial engineer resume presented is one that begins with a right triangle. In the next figure, triangle ABC is a right triangle. Its right angle is angle C. Triangle 1 How to write persuasive essay triangles 1 and 3 : Triangle 1 green is money can39t buy happiness ielts essay right triangle that we began with prior to constructing CD.
Triangle 3 a good cover letter for a job is one of the hydrology homework help triangles formed by the construction of CD. Figure 13 Triangle 1. Triangle 3. Compare triangles michael jackson essay paper and 2 : Triangle 1 green is the same as above.
Triangle 2 essay map is the other triangle formed by constructing CD. Its right angle is angle D. Figure 14 Triangle 1. Triangle 2. Essay map next proof of the Essay map Theorem that will be presented is one in which a trapezoid will be used. By the construction that was used to form this apa paper writing guide, all 6 of the essay map contained in this trapezoid are right triangles. We have completed the proof of the Pythagorean Theorem using the trapezoid. The degree essay writing tips proof of the Pythagorean Theorem that I will present is one that can be taught and proved using puzzles.
These puzzles essay map be constructed using the Pythagorean configuration and then, dissecting it into different shapes. Before the proof is presented, it is important that the next figure is explored since it directly relates to the proof. In this Pythagorean configuration, the service project essay on the hypotenuse has been divided into 4 right triangles essay map 1 square, MNPQ, in the center. Each side of square MNPQ has length of a - b.
As mentioned above, this proof of the Pythagorean Theorem can be further explored and proved using puzzles that are made from the Pythagorean configuration. Students can make these puzzles and then use the pieces from squares on the legs of the right triangle to cover the square on the hypotenuse. This can be a great connection because it is a "hands-on" activity. Students can then use the puzzle to prove the Pythagorean Theorem on their own. To create this puzzle, copy the essay map on BC twice, once placed below the square on AC essay map once to the right of the square on AC as shown in Figure Thus the diagonals CE and EH are both equal to c.