{"id":25,"date":"2022-02-17T00:16:22","date_gmt":"2022-02-17T00:16:22","guid":{"rendered":"https:\/\/convexabacus.xyz\/?p=25"},"modified":"2022-02-17T21:13:42","modified_gmt":"2022-02-17T21:13:42","slug":"golden-rectangles-in-a-right-triangle","status":"publish","type":"post","link":"https:\/\/convexabacus.xyz\/index.php\/2022\/02\/17\/golden-rectangles-in-a-right-triangle\/","title":{"rendered":"Golden Rectangles in a Right Triangle"},"content":{"rendered":"\n<p>This construction was created by experimenting with the Geogebra interactive geometry application. I began with the right-triangle with side lengths 3, 4, and 5. Then I constructed its incircle, which has unit radius. In each of the three corners I inscribed a smaller circle, tangent to the incircle. Considering the circle inscribed in the 3-5 corner, I observed that the numerical value of its radius is 1\/\u03c6<sup>2<\/sup>, where \u03c6 (phi) is the golden ratio, equal to (1+\u221a5)\/2. This enables construction of three golden rectangles as shown. The dimensions of the rectangles are 1\/\u03c6<sup>2<\/sup> x 1\/\u03c6, 1\/\u03c6 x 1, and 1 x \u03c6. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"999\" height=\"1024\" src=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/golden-rectangles-in-a-right-triangle-999x1024.png\" alt=\"\" class=\"wp-image-26\" srcset=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/golden-rectangles-in-a-right-triangle-999x1024.png 999w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/golden-rectangles-in-a-right-triangle-293x300.png 293w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/golden-rectangles-in-a-right-triangle-768x787.png 768w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/golden-rectangles-in-a-right-triangle-1499x1536.png 1499w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/02\/golden-rectangles-in-a-right-triangle-1998x2048.png 1998w\" sizes=\"auto, (max-width: 999px) 100vw, 999px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>This construction was created by experimenting with the Geogebra interactive geometry application. I began with the right-triangle with side lengths 3, 4, and 5. Then 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":[8,5],"tags":[],"class_list":["post-25","post","type-post","status-publish","format-standard","hentry","category-geogebra","category-geometry"],"_links":{"self":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/25","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=25"}],"version-history":[{"count":3,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/25\/revisions"}],"predecessor-version":[{"id":37,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/25\/revisions\/37"}],"wp:attachment":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=25"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=25"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=25"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}