{"id":298,"date":"2014-06-14T12:00:41","date_gmt":"2014-06-14T19:00:41","guid":{"rendered":"http:\/\/mostlymental.com\/?p=298"},"modified":"2014-06-13T20:57:39","modified_gmt":"2014-06-14T03:57:39","slug":"pythagoras-week-day-7","status":"publish","type":"post","link":"http:\/\/mostlymental.com\/?p=298","title":{"rendered":"Pythagoras Week: Day 7"},"content":{"rendered":"<p>Hello everyone!<\/p>\n<p>Welcome back to Pythagoras Week.\u00a0 If you missed it, be sure to check it out from the <a href=\"http:\/\/mostlymental.com\/?p=243\">beginning<\/a>.\u00a0 Our final proof doesn't rely on any sort of fancy results like the last few.\u00a0 We just need a few copies of our triangle.<\/p>\n<p>&nbsp;<\/p>\n<p>Take three copies of a right triangle with sides <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_64f47382e7ddc46583bf6d2abedf4140.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Scale one by <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_0cc175b9c0f1b6a831c399e269772661.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, one by <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_92eb5ffee6ae2fec3ad71c777531578f.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, and the third by <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_4a8a08f09d37b73795649038408b5f33.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Place them together as shown below.<a href=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_7.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-299 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_7-300x141.png\" alt=\"pyth_proof_7\" width=\"300\" height=\"141\" srcset=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_7-300x141.png 300w, http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_7.png 345w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>The angles work out nicely, and it turns out we get a rectangle.\u00a0 The top edge has length <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_91dcd1eb2fc092f982e5e87b8685d280.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script> and the bottom edge has length <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_2796af5074a7f27ecccd3cd17e165d53.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Opposite sides of a rectangle are the same length, so we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_d459699ab031562a6c4fbb577db204bc.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<\/p>\n<p>&nbsp;<\/p>\n<p>That's it for Pythagoras week.\u00a0 I'll post a final wrap-up tomorrow.<\/p>\n<a class=\"synved-social-button synved-social-button-share synved-social-size-48 synved-social-resolution-single synved-social-provider-facebook nolightbox\" data-provider=\"facebook\" target=\"_blank\" rel=\"nofollow\" title=\"Share on Facebook\" 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Welcome back to Pythagoras Week.\u00a0 If you missed it, be sure to check it out from the beginning.\u00a0 Our final proof doesn't rely on any sort of fancy results like the last few.\u00a0 We just need a few copies of our triangle. &nbsp; Take three copies of a right triangle with sides .\u00a0 &hellip; <a href=\"http:\/\/mostlymental.com\/?p=298\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Pythagoras Week: Day 7<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,2],"tags":[],"class_list":["post-298","post","type-post","status-publish","format-standard","hentry","category-geometry","category-math"],"_links":{"self":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/298","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=298"}],"version-history":[{"count":1,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/298\/revisions"}],"predecessor-version":[{"id":300,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/298\/revisions\/300"}],"wp:attachment":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=298"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=298"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=298"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}