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Department of Mathematics Education J. Wilson, EMT The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, what is an essay?, a Greek seaport in Southern Italy. 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 that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were what is an essay? in their belief that any two lengths were integral multiples of some unit length.
Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret 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 hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.
Figure 1. According to what is an essay? Pythagorean Theorem, the sum of the areas of the two red squares, what is an essay?, squares A and B, is equal to the area of the blue square, square C. for a right triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse. Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras.
It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem. Therefore, the square on c is equal to the sum of the squares on a and b. Burton There are many other proofs of the Pythagorean Theorem. One came from the contemporary Chinese civilization found in the oldest extant Chinese text containing formal mathematical theories, the Arithmetic Classic of the Gnoman and the Circular Paths of Heaven.
The proof of the Pythagorean Theorem that was inspired by a figure in this book was included in the book Vijaganita, Root Calculationsby the Hindu mathematician Bhaskara, what is an essay?. 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 earliest problems in the theory of numbers known as the Pythgorean problem.
Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:. The formula that will generate all Pythagorean triples first appeared in Book 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 introduced to students during the middle school years. This theorem becomes increasingly important during the high school years. It is what is an essay? enough to merely state the algebraic formula for the Pythagorean Theorem.
Students need to see the geometric 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. and plugging numbers into the formula. The following is a variety of proofs of the Pythagorean Theorem including one by Euclid, what is an essay?. These proofs, along with manipulatives and technology, what is an essay?, can greatly 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 the 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 lies along the altitude to the right triangle ABC.
See Figure 3. Figure 3, what is an essay?. Next, Euclid showed that the area of rectangle HBDG is equal to the area of square what is an essay? BC and that the are of the rectangle HAJG is equal to the area of the square on AC. He proved these equalities using the concept of similarity. Triangles ABC, AHC, and CHB are similar. The similarity of triangles ABC and AHC means. or, as to be proved, the area of the rectangle HAJG is the same as the areaof the square on side AC.
In the same way, what is an essay?, triangles ABC and CHG are similar. 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 the Pythagorean Theorem. Thus, what is an essay?, he used the result that parallelograms are double the triangles with the same base and between the same parallels. Draw CJ and BE. The area of the rectangle AHGJ is double the area of triangle JAC, what is an essay?, and the area of square ACLE is double triangle BAE.
What is an essay? two triangles are congruent by SAS, what is an essay?. The same result follows in a similar manner for the other rectangle and square, what is an essay?. Katz, Click here for a GSP animation to illustrate this proof. The next three proofs are more easily seen proofs of the Pythagorean Theorem and would be what is an essay? for high school mathematics students.
In fact, these are proofs that students could be able to what is an essay? 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 up into right triangles.
Figure 4 Figure 5. It can be seen that triangles 2 in green and 1 in redwill completely overlap triangle 3 in blue. Now, we can 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. What is an essay? 8. We have proved the Pythagorean Theorem. The next proof is another proof of the Pythagorean Theorem that begins with a rectangle. Figure 9. By the AA similarity theorem, triangle EBF is similar to triangle CAB. Now, let k be the similarity ratio between triangles EBF and CAB. Figure Thus, triangle EBF has sides with lengths ka, kb, and kc. By solving for k, we have.
and we have completed the proof. The next proof of the Pythagorean Theorem that will be 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 Compare triangles 1 and 3 : Triangle 1 green is the right triangle that we began with prior to constructing CD. Triangle 3 red is one of the two triangles formed by the construction of CD.
Figure 13 Triangle 1. Triangle 3. Compare triangles 1 and 2 : Triangle 1 green is the same as above. Triangle 2 blue is the other triangle formed by constructing CD. Its right angle is angle What is an essay?. Figure 14 Triangle 1. Triangle 2, what is an essay?. The next proof of the Pythagorean Theorem that will be presented is one in which a trapezoid will be used. By the construction that was used to form this trapezoid, all 6 of the triangles contained in this trapezoid are right triangles.
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COMPARE AND CONTRAST ESSAY OUTLINE. I. Introduction. A. Hook: _____ _____ B An essay is a short nonfictional piece of formal writing assigned to students to improve their writing skills or assess their knowledge of a given subject. Alternative definitions: According to Frederick Crews, professor of English at the University of California at Berkeley, an essay is “a fairly brief piece of nonfiction that tries to make a point in an interesting way.”Estimated Reading Time: 7 mins The next three proofs are more easily seen proofs of the Pythagorean 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
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