{"id":47,"date":"2022-02-20T14:27:51","date_gmt":"2022-02-20T14:27:51","guid":{"rendered":"https:\/\/convexabacus.xyz\/?p=47"},"modified":"2022-03-24T22:52:19","modified_gmt":"2022-03-24T22:52:19","slug":"proof-that-the-golden-ratio-is-in-the-3-4-5-right-triangle","status":"publish","type":"post","link":"https:\/\/convexabacus.xyz\/index.php\/2022\/02\/20\/proof-that-the-golden-ratio-is-in-the-3-4-5-right-triangle\/","title":{"rendered":"Proof that the Golden Ratio is in the 3-4-5 Right Triangle"},"content":{"rendered":"\n<p>In my previous post <a href=\"https:\/\/convexabacus.xyz\/index.php\/2022\/02\/17\/golden-rectangles-in-a-right-triangle\/\">Golden Rectangles in a Right Triangle<\/a> I inscribed circles in a right-angled triangle with sides of length 3, 4, and 5. I measured that the radius of one inscribed circle was 1\/\u03c6<sup>2<\/sup>. This was an empirical observation of limited accuracy, not a mathematical proof. Phi or \u03c6, equal to (1+\u221a5)\/2, is a transcendental number. <em>Note Paul&#8217;s correction in the comments : phi is not transcendental<\/em>. Fortunately, my colleague Ed Staples was able to prove the length of the radius using trigonometry. His work is shown in the figure. The length <em>r<\/em> to be calculated is the radius of the small circle inscribed in the corner between the sides of length 3 and length 5. He proves that <em>r<\/em> = 2 \u2013 \u03c6. And, 2 \u2013 \u03c6 is equal to 1\/\u03c6<sup>2<\/sup>, which is the value that was determined approximately from the construction. I used the Geogebra software to make this construction. The software conveniently measures lengths and angles, which was particularly useful in this case. You can find Ed Staples\u2019 blog, with a variety of mathematical puzzles and tutorials, at <a href=\"https:\/\/www.mathematicalwhetstones.com\/blog\">https:\/\/www.mathematicalwhetstones.com\/blog<\/a><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"811\" height=\"1024\" src=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/proof-that-phi-is-in-the-3-4-5-right-triangle-by-ed-staples-811x1024.jpg\" alt=\"\" class=\"wp-image-48\" srcset=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/proof-that-phi-is-in-the-3-4-5-right-triangle-by-ed-staples-811x1024.jpg 811w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/proof-that-phi-is-in-the-3-4-5-right-triangle-by-ed-staples-237x300.jpg 237w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/proof-that-phi-is-in-the-3-4-5-right-triangle-by-ed-staples-768x970.jpg 768w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/proof-that-phi-is-in-the-3-4-5-right-triangle-by-ed-staples-1216x1536.jpg 1216w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/proof-that-phi-is-in-the-3-4-5-right-triangle-by-ed-staples.jpg 1280w\" sizes=\"auto, (max-width: 811px) 100vw, 811px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>In my previous post Golden Rectangles in a Right Triangle I inscribed circles in a right-angled triangle with sides of length 3, 4, and 5. I&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,9],"tags":[],"class_list":["post-47","post","type-post","status-publish","format-standard","hentry","category-geometry","category-trigonometry"],"_links":{"self":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=47"}],"version-history":[{"count":3,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/47\/revisions"}],"predecessor-version":[{"id":74,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/47\/revisions\/74"}],"wp:attachment":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=47"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=47"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}