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 the 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 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. 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. 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. The Pythagorean Theorem was known to the Babylonians about years before Pythagoras was born, though Pythagoras is credited with the first recorded proof. Though we name this theorem after the Greek mathematician Pythagoras who lived approximately BC, it was known and recorded by the Babylonian mathematicians about years earlier.
It was also discovered independently by other ancient civilisations in Mesopotamia, India and China. In what year was the Pythagorean Theorem developed? George C. Nov 22, The Egyptians wanted a perfect degree angle to build the pyramids which were actually two right-angle triangle whose hypotenuse forms the edges of the pyramids. There are some clues that the Chinese had also developed the Pythagoras theorem using the areas of the sides long before Pythagoras himself.
But they did not actually write them down and so Pythagoras gets the credit for simply writing them down. Pythagoras was born in around BC, in an island called Samos in Greece. There is no much information about his youth though he did a lot of travelling to study is all that is known. Latter Pythagoras settled in Crotone a city and comune in Calabria , where he started his cult or group called the Pythagoreans.
The Pythagoreans loved maths so much that it was like a god to them, they pretty much worshiped maths. They believed that numbers ruled the universe with its mystical and spiritual qualities.
But when this theorem was discovered and proved the Pythagorean sacrifice huge number of bulls to their number gods. Indian mathematicians in the ancient times knew the Pythagorean theorem, they also used something called the Sulbasutras that discuss the theorem in the context of strict requirements for the orientation, shape, and area of altars for religious purposes.
The Pythagorean theorem was first originated in ancient Babylon and Egypt beginning about B. It sure is amazing to know such a story behind such a simple proof of Pythagoras theorem. To know more of such amazing stories stay with us on Embibe.
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