{"id":2443,"date":"2020-05-24T13:32:39","date_gmt":"2020-05-24T11:32:39","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2443"},"modified":"2021-10-26T16:39:25","modified_gmt":"2021-10-26T14:39:25","slug":"un-probleme-abordable-au-college","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/24\/un-probleme-abordable-au-college\/","title":{"rendered":"Un probl\u00e8me abordable au coll\u00e8ge"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">La question est simple : quelle est la longueur AE dans la figure suivante ?<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"472\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/01.png\" alt=\"\" class=\"wp-image-2444\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/01.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/01-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/01-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/01-150x150.png 150w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><figcaption>Que vaut AE ?<\/figcaption><\/figure><\/div>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">Nous savons que ABCD est un carr\u00e9 et que (AF) est la bissectrice de \\(\\widehat{DAE}\\). Nous avons not\u00e9 \\(x = AD\\) la longueur d&rsquo;un c\u00f4t\u00e9 de ce carr\u00e9, mais nous n&rsquo;en avons pas n\u00e9cessairement besoin.<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_87 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/24\/un-probleme-abordable-au-college\/#1ere_approche_abordable_au_college\" >1\u00e8re approche : abordable au coll\u00e8ge<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/24\/un-probleme-abordable-au-college\/#Autres_approches\" >Autres approches<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"1ere_approche_abordable_au_college\"><\/span>1\u00e8re approche : abordable au coll\u00e8ge<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"472\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/02.png\" alt=\"\" class=\"wp-image-2445\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/02.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/02-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/02-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/02-150x150.png 150w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Notons \\(\\theta = \\widehat{DAF} = \\widehat{FAE}\\) (car (AF) est la bissectrice de \\(\\widehat{DAE}\\)).<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"472\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/03.png\" alt=\"\" class=\"wp-image-2446\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/03.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/03-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/03-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/03-150x150.png 150w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Le triangle FAD est rectangle en D donc les angles \\(\\widehat{FAD}\\) et \\(\\widehat{DFA}\\) sont compl\u00e9mentaires:$$\\widehat{DFA}=90-\\widehat{FAD}=90-\\theta.$$On convient bien entendu que tous les angles sont exprim\u00e9s en degr\u00e9s.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"472\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/04.png\" alt=\"\" class=\"wp-image-2447\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/04.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/04-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/04-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/04-150x150.png 150w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Les angles \\(\\widehat{DAE}\\) et \\(\\widehat{EAB}\\) sont adjacents et compl\u00e9mentaires donc: $$\\widehat{EAB}=90-\\widehat{DAE}=90-2\\theta.$$ De plus, \\(\\widehat{BEA}=\\widehat{DAE}\\) car ils sont alternes-internes form\u00e9s par deux droites parall\u00e8les : (AD) et (BC).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Faisons donc une rotation de centre A et d&rsquo;angle \\(2\\theta\\) dans le sens des aiguilles d&rsquo;une montre:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"629\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/05.gif\" alt=\"\" class=\"wp-image-2448\"\/><figcaption>Rotation de \\(2\\theta\\)<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">On obtient:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"596\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/06.png\" alt=\"\" class=\"wp-image-2449\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/06.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/06-300x379.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/06-238x300.png 238w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Notons E&rsquo; l&rsquo;image de E par cette rotation. Si l&rsquo;on regarde l&rsquo;angle \\(\\widehat{E&rsquo;AF}\\), sa mesure vaut \\(90-2\\theta+\\theta=90-\\theta\\):<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"599\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/07.png\" alt=\"\" class=\"wp-image-2450\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/07.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/07-300x381.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/07-236x300.png 236w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, dans le triangle E&rsquo;AF, les angles \\(\\widehat{E&rsquo;AF}\\) et \\(\\widehat{E&rsquo;FA}\\) ont la m\u00eame mesure:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"472\" height=\"590\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/08.png\" alt=\"\" class=\"wp-image-2451\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/08.png 472w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/08-300x375.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/08-240x300.png 240w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ce triangle est donc isoc\u00e8le en E&rsquo;; ainsi E&rsquo;A = E&rsquo;F.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or, E&rsquo;A = EA car une rotation conserve les longueurs (rappelons que [E&rsquo;A] est l&rsquo;image de [EA] par la rotation consid\u00e9r\u00e9e).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, EA = E&rsquo;A = E&rsquo;F = 36 + 64 = 100.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Autres_approches\"><\/span>Autres approches<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Ce probl\u00e8me est expos\u00e9 dans la vid\u00e9o de la cha\u00eene <em>MindYourDecisions<\/em> suivante, o\u00f9 l&rsquo;on y expose d&rsquo;autres approches plus compliqu\u00e9es, dont une qui m\u00e8ne \u00e0 une \u00e9quation de degr\u00e9 2, mais en faisant intervenir des tangentes et une relation sur les tangentes qui n&rsquo;est pas un programme du lyc\u00e9e en France.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Think Outside Or Inside The Box!\" width=\"750\" height=\"422\" src=\"https:\/\/www.youtube.com\/embed\/jsajsxV4icI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Les fichiers sources \\(\\LaTeX\\) des figures sont disponibles ci-dessous pour les abonn\u00e9\u00b7e\u00b7s de mathweb.fr :<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/article-2020-05-24.zip\">Fichier ZIP<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/article-2020-05-24.zip\" class=\"wp-block-file__button\" download=\"\">T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>La question est simple : quelle est la longueur AE dans la figure suivante ?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[59,52],"class_list":["post-2443","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-angles","tag-geometrie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Un probl\u00e8me abordable au coll\u00e8ge - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Dans un carr\u00e9 ABCD, E est un point de [BC] tel que BE = 36. 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