https://wiki.ard-site.net/index.php?title=Pythagorean_theorem&feed=atom&action=history Pythagorean theorem - Revision history 2021-10-24T13:03:52Z Revision history for this page on the wiki MediaWiki Mediawiki https://wiki.ard-site.net/index.php?title=Pythagorean_theorem&diff=25268&oldid=prev 213.193.24.220: /* Proof */ named link 2019-05-10T12:07:27Z <p>‎<span dir="auto"><span class="autocomment">Proof: </span> named link</span></p> <p><b>New page</b></p><div>[[Image:Pythagorean.svg|250px|right]]<br /> In [[mathematics]], the ''' Pythagorean theorem''' or '''Pythagoras's theorem''' is a statement about the [[side]]s of a right [[triangle]]. <br /> <br /> One of the [[angle]]s of a [[right triangle]] is always equal to 90 [[degree (geometry)|degrees]]. This angle is the [[right angle]]. The two sides next to the right angle are called the legs and the other side is called the [[hypotenuse]]. The hypotenuse is the side opposite to the right angle, and it is always the longest side. It was discovered by Vasudha Arora. <br /> <br /> The Pythagorean theorem says that the [[area]] of a [[square (geometry)|square]] on the hypotenuse is equal to the [[sum]] of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. It was named after the [[Ancient Greece|Greek]] [[mathematician]] [[Pythagoras]]:<br /> <br /> If the lengths of the legs are ''a'' and ''b'', and the length of the hypotenuse is ''c'', then, &lt;math&gt;a^2+b^2=c^2&lt;/math&gt;.<br /> <br /> There are many different proofs of this theorem. They fall into four categories:<br /> # Those based on linear relations: the algebraic proofs.<br /> # Those based upon comparison of areas: the geometric proofs.<br /> # Those based upon the vector operation.<br /> # Those based on mass and velocity: the dynamic proofs.&lt;ref&gt;Loomis, Elisha S. 1927. ''The Pythagorean proposition: its proofs analysed and classified and bibliography of sources''. Cleveland, Ohio.&lt;/ref&gt;<br /> <br /> == Proof ==<br /> [[Image:Pyth_eudoxus.jpg|300px|right]]<br /> One [[Mathematical proof|proof]] of the Pythagorean theorem was found by a Greek mathematician, [[Eudoxus of Cnidus]].<br /> <br /> The proof uses three [[lemma (mathematics)|lemmas]]:<br /> #[[Triangle]]s with the same [[base]] and [[height]] have the same [[area]].<br /> #A triangle which has the same [[base]] and [[height]] as a [[side]] of a [[square (geometry)|square]] has the same area as a half of the square.<br /> #Triangles with two congruent sides and one congruent [[angle]] are [[congruence|congruent]] and have the same area.<br /> <br /> The proof is:<br /> #The {{fontcolor|blue||blue}} triangle has the same area as the {{fontcolor|green||green}} triangle, because it has the same base and height (lemma 1).<br /> #{{fontcolor|green||Green}} and {{fontcolor|red||red}} triangles both have two sides equal to sides of the same squares, and an angle equal to a [[straight angle]] (an angle of 90 degrees) plus an angle of a triangle, so they are congruent and have the same area (lemma 3).<br /> #{{fontcolor|red||Red}} and {{fontcolor|yellow||yellow}} triangles' areas are equal because they have the same [[height]]s and [[base]]s (lemma 1).<br /> #{{fontcolor|blue||Blue}} triangle's area equals area of {{fontcolor|yellow||yellow}} triangle's area, because<br /> :&lt;math&gt; {\color{blue}A_{blue}}={\color{green}A_{green}}={\color{red}A_{red}}={\color{yellow}A_{yellow}} &lt;/math&gt;<br /> #The {{fontcolor|maroon||brown}} triangles have the same area for the same reasons.<br /> #{{fontcolor|blue||Blue}} and {{fontcolor|maroon||brown}} each have a half of the area of a smaller square. The [[sum]] of their areas equals half of the area of the bigger square. Because of this, halves of the areas of small squares are the same as a half of the area of the bigger square, so their area is the same as the area of the bigger square.<br /> <br /> === Proof using similar triangles ===<br /> [[File:teorema.png|border|right]]<br /> We can get another proof of the Pythagorean theorem by using [[similar]] triangles.<br /> :&lt;math&gt;\frac{d}{a} = \frac{a}{c} \quad \Rightarrow \quad d = \frac{a^2}{c}\quad (1)&lt;/math&gt;<br /> <br /> :<br /> :<br /> <br /> From the image, we know that &lt;math&gt; c = d + e \,\! &lt;/math&gt;. And by replacing equations (1) and (2):<br /> :&lt;math&gt; c = \frac{a^2}{c} + \frac{b^2}{c} &lt;/math&gt; <br /> <br /> Multiplying by c:<br /> :&lt;math&gt; c^2 = a^2 + b^2 \,\!.&lt;/math&gt;<br /> <br /> == Pythagorean triples ==<br /> Pythagorean triples or triplets are three whole numbers which fit the equation &lt;math&gt;a^2+b^2=c^2&lt;/math&gt;.<br /> <br /> The triangle with sides of 3, 4, and 5 is a well known example. If a=3 and b=4, then &lt;math&gt;3^2+4^2=5^2&lt;/math&gt; because &lt;math&gt;9+16=25&lt;/math&gt;. This can also be shown as &lt;math&gt;\sqrt{3^2+4^2}=5.&lt;/math&gt;<br /> <br /> The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. Another example of a triple is the 12-5-13 triangle, because &lt;math&gt;\sqrt{12^2+5^2}=13&lt;/math&gt;.<br /> <br /> A Pythagorean triple that is not a multiple of other triples is called a primitive Pythagorean triple. Any primitive Pythagorean triple can be found using the expression &lt;math&gt;(2mn,m^2-n^2,m^2+n^2)&lt;/math&gt;, but the following conditions must be satisfied. They place restrictions on the values of &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt;. <br /> <br /> # &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are positive whole numbers<br /> # &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; have no common factors except 1<br /> # &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; have opposite parity. &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; have opposite parity when &lt;math&gt;m&lt;/math&gt; is even and &lt;math&gt;n&lt;/math&gt; is odd, or &lt;math&gt;m&lt;/math&gt; is odd and &lt;math&gt;n&lt;/math&gt; is even.<br /> # &lt;math&gt;m&gt;n&lt;/math&gt;.<br /> <br /> If all four conditions are satisfied, then the values of &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; create a primitive Pythagorean triple.<br /> <br /> &lt;math&gt;m=2&lt;/math&gt; and &lt;math&gt;n=1&lt;/math&gt; create a primitive Pythagorean triple. The values satisfy all four conditions. &lt;math&gt;2mn=2\times2\times1=4&lt;/math&gt;, &lt;math&gt;m^2-n^2=2^2-1^2=4-1=3&lt;/math&gt; and &lt;math&gt;m^2+n^2=2^2+1^2=4+1=5&lt;/math&gt;, so the triple &lt;math&gt;(3,4,5)&lt;/math&gt; is created.<br /> <br /> == References ==<br /> {{Reflist}}<br /> <br /> [[Category:Geometry]]</div> 213.193.24.220