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 in a similar manner for the other rectangle and square. 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 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 up into right triangles. Figure 4 Figure 5. It can be seen that triangles 2 in green and 1 in red , will completely overlap triangle 3 in blue.
Now, we can give a proof of the Pythagorean Theorem using these same triangles. Compare triangles 1 and 3. Angles E and D, respectively, are the right angles in these triangles. By comparing their similarities, we have. We have proved the Pythagorean Theorem. The next proof is another proof of the Pythagorean Theorem that begins with a rectangle. Thus, triangle EBF has sides with lengths ka, kb, and kc. By solving for k, we have. 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. 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 D.
Figure 14 Triangle 1. 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. We have completed the proof of the Pythagorean Theorem using the trapezoid. The next proof of the Pythagorean Theorem that I will present is one that can be taught and proved using puzzles.
These puzzles can 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 square on the hypotenuse has been divided into 4 right triangles and 1 square, MNPQ, in the center.
Each side of square MNPQ has length of a - b. This gives the following: 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 square on BC twice, once placed below the square on AC and once to the right of the square on AC as shown in Figure Proof Using Figure Thus the diagonals CE and EH are both equal to c. Pieces 4 and 7, and pieces 5 and 6 are not separated. By calculating the area of each piece, it can be shown that. So shocked were the Pythagoreans by these numbers, they put to death a member who dared to mention their existence to the public.
It would be years later that the Greek mathematician Eudoxus developed a way to deal with these unutterable numbers. Pythagoras of Samos Who is Pythagoras? He was born to Mnesarchus and Pythais. He was one of either three or four children, there is proof for both of these accounts. At a very early age, Pythagoras learned to play the lyre and recite Homer.
Pythagoras also wrote poetry at a very early age. Pythagoras got his education from three philosophers, the most important mathematically being Anaximander. Around BC Pythagoras went on a journey to Croton, where he established a philosophical and religious school.
Pythagoras was the head of the inner circle of the society, his inside followers were called mathematikoi. The mathematikoi had to follow very strict laws that Pythagoras believed. The five things that Pythagoras believed very deeply were 1 that at its deepest level, reality is mathematical in nature, 2 that philosophy can be used for spiritual purification, 3 that the soul can rise to union with the divine, 4 that certain symbols have a mystical significance, and 5 that all brothers of the order should observe strict loyalty and secrecy.
Both men and women were allowed to join the society. Pythagoras was not acting as a modern research group at a major university.
The Pythagorean theorem states that: "The area of the square built on the hypotenuse of a right triangle is equal to the sum of the squares on the remaining two sides." According to the Pythagorean Theorem, the sum of the areas of the red and yellow squares is equal to the area of the purple square.
One of the topics that almost every high school geometry student learns about is the Pythagorean Theorem. When asked what the Pythagorean Theorem is, students will often state that a2+b2=c2 where a, b, and c are sides of a right triangle.
Fermat's Last Theorem - Rationale: The pythagorean theorem is a simple equation that has been taught to pupils from the beginning of middle school. a2+b2=c2 is the basic formula to calculate any one of the sides on a right angle triangle. The Pythagorean Theorem Essay Sample Introduction a 2 + b 2 = c 2 where a and b are the sides of a right triangle and c is the hypotenuse is the answer most students will give when asked to define the Pythagorean Theorem.
One theorem that is particularly renowned is the Pythagorean Theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of any right triangle. Essay, Research Paper: Pythagorean Theorem Mathematics Free Mathematics research papers were donated by our members/visitors and are presented free of charge for informational use only.