{"id":251,"date":"2014-06-09T12:00:46","date_gmt":"2014-06-09T19:00:46","guid":{"rendered":"http:\/\/mostlymental.com\/?p=251"},"modified":"2014-06-08T03:13:51","modified_gmt":"2014-06-08T10:13:51","slug":"pythagoras-week-day-2","status":"publish","type":"post","link":"http:\/\/mostlymental.com\/?p=251","title":{"rendered":"Pythagoras Week: Day 2"},"content":{"rendered":"<p>Hello everyone!<\/p>\n<p>Welcome back to Pythagoras Week.\u00a0 If you missed <a href=\"http:\/\/mostlymental.com\/?p=243\">yesterday's post<\/a>, be sure to check it out.\u00a0 Today, I'd like to show you a proof originally given by President James Garfield.<\/p>\n<p>Suppose once again we have 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 Construct a trapezoid with the dimensions shown below on the left.\u00a0 As you might recall, the area of a trapezoid is half the sum of the bases times the height.\u00a0 Thus, the area of the trapezoid on the left is <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_3fb14e1852993e49ba0522a41a89999d.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Next, break this trapezoid into three right triangles, as show at the right.\u00a0 The area of a triangle is half the base times the height.\u00a0 Thus, the total area of the three triangles is <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_59b1399ac2bdffa37db5fcbb88cfae96.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.<a href=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-255 aligncenter\" src=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_2-300x185.png\" alt=\"pyth_proof_2\" width=\"300\" height=\"185\" srcset=\"http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_2-300x185.png 300w, http:\/\/mostlymental.com\/wp-content\/uploads\/2014\/06\/pyth_proof_2.png 392w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>We've got the same trapezoid, so the two areas must be the same.\u00a0 Thus, <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_e94e4f8d5c0a1cb4abea26c162759955.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Multiplying both sides by <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_c81e728d9d4c2f636f067f89cc14862c.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>, we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_9114a718a265c7c226766eeb59f3c683.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 With a little algebra, we get <span class='MathJax_Preview'><img src='http:\/\/mostlymental.com\/wp-content\/plugins\/latex\/cache\/tex_f4f1aef6f7d63959e769c424b32adbf5.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\" \/><\/span><script type='math\/tex'><\/script>.\u00a0 Thus, <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 today.\u00a0 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Welcome back to Pythagoras Week.\u00a0 If you missed yesterday's post, be sure to check it out.\u00a0 Today, I'd like to show you a proof originally given by President James Garfield. Suppose once again we have a right triangle with sides .\u00a0 Construct a trapezoid with the dimensions shown below on the left.\u00a0 As &hellip; <a href=\"http:\/\/mostlymental.com\/?p=251\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Pythagoras Week: Day 2<\/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-251","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\/251","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=251"}],"version-history":[{"count":2,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/251\/revisions"}],"predecessor-version":[{"id":256,"href":"http:\/\/mostlymental.com\/index.php?rest_route=\/wp\/v2\/posts\/251\/revisions\/256"}],"wp:attachment":[{"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=251"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=251"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mostlymental.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}