{"id":94,"date":"2022-04-12T18:57:57","date_gmt":"2022-04-12T18:57:57","guid":{"rendered":"https:\/\/convexabacus.xyz\/?p=94"},"modified":"2022-04-12T18:58:46","modified_gmt":"2022-04-12T18:58:46","slug":"two-equal-rectangles-in-a-semicircle","status":"publish","type":"post","link":"https:\/\/convexabacus.xyz\/index.php\/2022\/04\/12\/two-equal-rectangles-in-a-semicircle\/","title":{"rendered":"Two Equal Rectangles in a Semicircle"},"content":{"rendered":"\n<p>In this construction two equal rectangles are jammed into a semicircle, one sitting vertically and the other leaning over at a tilt of 30 degrees. The puzzle is described as follows: Two equal rectangles, ABCD and STUV, both have a width 1 but an unknown height <em>h<\/em>. The left rectangle sits on the diameter of a circle of radius <em>r<\/em> and has its top left vertex B touching it as shown in the diagram. The right rectangle, leaning at 30 degrees to the diameter, touches the circle at T and U, has its bottom left vertex S touching the left rectangle, and its bottom right vertex V touching the diameter shown. Can you calculate exact expressions for the height <em>h<\/em> of the rectangles and the radius <em>r<\/em> of the circle? I thank Ed Staples for this concise statement of my problem, for his diagram shown below, and for his solution which will be shared later. ( Permit a little history. In my first draft the rectangles were of fixed dimension 2 x 1 and the tilt angle was unknown. Unable to solve this, I fixed the tilt angle at 30 degrees and made the height unknown &#8211; resulting in nicer problem. )<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"750\" height=\"653\" src=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/04\/Problem_two_rectangles_in_semicircle_stated_by_Ed_Staples.jpeg\" alt=\"\" class=\"wp-image-95\" srcset=\"https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/04\/Problem_two_rectangles_in_semicircle_stated_by_Ed_Staples.jpeg 750w, https:\/\/convexabacus.xyz\/wp-content\/uploads\/2022\/04\/Problem_two_rectangles_in_semicircle_stated_by_Ed_Staples-300x261.jpeg 300w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>In this construction two equal rectangles are jammed into a semicircle, one sitting vertically and the other leaning over at a tilt of 30 degrees. The&#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],"tags":[],"class_list":["post-94","post","type-post","status-publish","format-standard","hentry","category-geogebra"],"_links":{"self":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/94","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=94"}],"version-history":[{"count":1,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/94\/revisions"}],"predecessor-version":[{"id":96,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/posts\/94\/revisions\/96"}],"wp:attachment":[{"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=94"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=94"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/convexabacus.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}