{"id":13818,"date":"2016-01-09T08:33:13","date_gmt":"2016-01-09T08:33:13","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=13818"},"modified":"2018-02-13T16:14:37","modified_gmt":"2018-02-13T16:14:37","slug":"math-are-there-integer-pythagorean-triples-with-two-equal-sides","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-are-there-integer-pythagorean-triples-with-two-equal-sides\/","title":{"rendered":"Math:  Are there (integer) Pythagorean triples with two equal sides?"},"content":{"rendered":"<h1>Continuing about Pythagorean triples, the tutor considers the isosceles case.<\/h1>\n<p>In my <a href=\"?p=13806\">previous post<\/a> I discussed all-integer solutions to the Pythagoras equation<\/p>\n<p>a^2 + b^2 =c^2<\/p>\n<p>Such solutions are often called Pythagorean triples.<\/p>\n<p>Presently we consider the possibility of Pythagorean triples that have two equal numbers &#8211; that is, integer solutions to<\/p>\n<p>x^2 +x^2 = y^2<\/p>\n<p>The equation simplifies to <\/p>\n<p>2x^2 = y^2<\/p>\n<p>Dividing both sides by x^2 gives<\/p>\n<p>2=y^2\/x^2<\/p>\n<p>Square rooting both sides, we arrive at<\/p>\n<p>2^(1\/2)=y\/x<\/p>\n<p>Since 2^(1\/2) is not a rational number,<\/p>\n<p>2^(1\/2)=y\/x<\/p>\n<p>has no integer solutions for both x and y.<\/p>\n<p>Therefore, there are no (all-integer) Pythagorean triples of the form x,x,y.<\/p>\n<p>Source:<\/p>\n<p>Dudley, Underwood.  <em>Elementary Number Theory<\/em>, 2nd Ed.  New York:  W H Freeman<br \/>\u00a0\u00a0 and Company, 1978.<\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Continuing about Pythagorean triples, the tutor considers the isosceles case. In my previous post I discussed all-integer solutions to the Pythagoras equation a^2 + b^2 =c^2 Such solutions are often called Pythagorean triples. Presently we consider the possibility of Pythagorean &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/math-are-there-integer-pythagorean-triples-with-two-equal-sides\/\"> <span class=\"screen-reader-text\">Math:  Are there (integer) Pythagorean triples with two equal sides?<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[1328,1330,1331],"class_list":["post-13818","post","type-post","status-publish","format-standard","hentry","category-math","tag-pythagorean-triples","tag-right-isosceles-triangles","tag-sqaure-root-of-2-not-a-rational-number"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/13818","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=13818"}],"version-history":[{"count":15,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/13818\/revisions"}],"predecessor-version":[{"id":29609,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/13818\/revisions\/29609"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=13818"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=13818"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=13818"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}