{"id":9470,"date":"2024-01-18T17:21:43","date_gmt":"2024-01-18T16:21:43","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9470"},"modified":"2024-01-18T17:23:25","modified_gmt":"2024-01-18T16:23:25","slug":"un-probleme-mathematique-daire","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/","title":{"rendered":"Un probl\u00e8me math\u00e9matique d&rsquo;aire"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Il y a quelques temps, j&rsquo;ai vu le probl\u00e8me math\u00e9matique d&rsquo;aire suivant: \u00e9tant donn\u00e9e la figure suivante, trouver l&rsquo;aire manquante.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"298\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-298x300.png\" alt=\"probl\u00e8me math\u00e9matique aire\" class=\"wp-image-9471\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-298x300.png 298w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-160x160.png 160w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-600x604.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-1017x1024.png 1017w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-150x150.png 150w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire-768x774.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire.png 1513w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/a><\/figure>\n<\/div>\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 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\/2024\/01\/18\/un-probleme-mathematique-daire\/#Probleme_mathematique_daire_premiere_approche\" >Probl\u00e8me math\u00e9matique d&rsquo;aire: premi\u00e8re approche<\/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\/2024\/01\/18\/un-probleme-mathematique-daire\/#Probleme_mathematique_daire_une_approche_plus_algebrique\" >Probl\u00e8me math\u00e9matique d&rsquo;aire: une approche plus alg\u00e9brique<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/#Probleme_mathematique_Aire_de_AEIG\" >Probl\u00e8me math\u00e9matique: Aire de AEIG<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/#Aire_de_EBFI\" >Aire de EBFI<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/#Aire_de_HCFI\" >Aire de HCFI<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/#Aire_de_GDHI\" >Aire de GDHI<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/#Egalites\" >\u00c9galit\u00e9s<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/18\/un-probleme-mathematique-daire\/#Conclusion\" >Conclusion<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Probleme_mathematique_daire_premiere_approche\"><\/span>Probl\u00e8me math\u00e9matique d&rsquo;aire: premi\u00e8re approche<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Je vais noter \\(c\\) la mesure du carr\u00e9. Ainsi, la somme des quatre aires est \u00e9gale \u00e0 \\( c^2\\), donc:$$c^2=17+x\\quad\\text{soit}\\quad c=\\sqrt{17+x}.$$ On peut raisonnablement accepter le fait que l&rsquo;aire du carr\u00e9 ne d\u00e9passe pas 36, et donc que \\(c\\) ne soit pas plus grand que 6. Mais la donn\u00e9e de cette fonction de \\(x\\) ne nous avance pas trop&#8230;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"249\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-300x249.png\" alt=\"\" class=\"wp-image-9472\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-300x249.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-600x499.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image.png 682w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption class=\"wp-element-caption\">Courbe repr\u00e9sentative de la fonction \\(x \\mapsto \\sqrt{17+x} \\) sur [0;6]<\/figcaption><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Probleme_mathematique_daire_une_approche_plus_algebrique\"><\/span>Probl\u00e8me math\u00e9matique d&rsquo;aire: une approche plus alg\u00e9brique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cette fois-ci, je vais noter \\(c\\) la mesure de la moiti\u00e9 d&rsquo;un c\u00f4t\u00e9 du carr\u00e9, et je vais introduire des variables \\(y\\) et \\(z\\) comme indiqu\u00e9 ci-dessous sur la figure:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire3.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"285\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire3-300x285.png\" alt=\"\" class=\"wp-image-9474\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire3-300x285.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire3-600x570.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/aire3.png 747w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Probleme_mathematique_Aire_de_AEIG\"><\/span>Probl\u00e8me math\u00e9matique: Aire de AEIG<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">C&rsquo;est la somme de l&rsquo;aire des triangles AEI et AIG:$$\\mathcal{A}_1=\\frac{1}{2}c(2c-z)  + \\frac{2}{2}cy = \\frac{1}{2}c(2c-z+y)=6.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aire_de_EBFI\"><\/span>Aire de EBFI<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">C&rsquo;est la somme de l&rsquo;aire des triangles EBI et BIF:$$\\mathcal{A}_2=\\frac{1}{2}c(2c-z+2c-y) = \\frac{1}{2}c(4c-z-y)=4.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aire_de_HCFI\"><\/span>Aire de HCFI<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">C&rsquo;est la somme de l&rsquo;aire des triangles CFI et HCI:$$\\mathcal{A}_3=\\frac{1}{2}c(2c-y+z)=7.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aire_de_GDHI\"><\/span>Aire de GDHI<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">C&rsquo;est la somme de l&rsquo;aire des triangles GDI et DHI:$$\\mathcal{A_4}=\\frac{1}{2}c(y+z)=x.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Egalites\"><\/span>\u00c9galit\u00e9s <span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">De tout ceci, on en d\u00e9duit notamment les \u00e9galit\u00e9s:$$\\begin{cases}c(2c-z)=12\\\\c(4c-z-y)=8\\\\c(2c-y+z)=14\\\\c(y+z)=2x\\end{cases}$$Des deuxi\u00e8me et quatri\u00e8me \u00e9galit\u00e9s, on d\u00e9duit:$$c(4c-(y+z))=8 \\iff 4c^2-2x=8 \\iff x = 2c^2-4.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Comme nous l&rsquo;avons mentionn\u00e9 plus haut, on peut raisonnablement penser que \\( 6 &lt; x &lt; 36\\) d&rsquo;o\u00f9:$$\\begin{align}6 &lt; 2c^2-4 &lt; 36 &amp; \\iff 3 &lt; c^2-2 &lt; 18\\\\ &amp; \\iff 5 &lt; c^2 &lt; 20\\\\&amp;\\iff \\sqrt3 &lt; c &lt; 2\\sqrt5 \\end{align}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On sait que:$$\\begin{cases}x = 2c^2-4\\\\4c^2=17+x\\end{cases}$$et donc:$$2c^2-4=4x^2-17 \\iff 2c^2=13.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On en d\u00e9duit alors que:$$x=9.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Il y a quelques temps, j&rsquo;ai vu le probl\u00e8me math\u00e9matique d&rsquo;aire suivant: \u00e9tant donn\u00e9e la figure suivante, trouver l&rsquo;aire manquante.<\/p>\n","protected":false},"author":1,"featured_media":9480,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[54],"class_list":["post-9470","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-python","tag-aire"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Un probl\u00e8me math\u00e9matique d&#039;aire - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Il y a quelques temps, j&#039;ai vu le probl\u00e8me math\u00e9matique d&#039;aire suivant: \u00e9tant donn\u00e9e la figure suivante, trouver 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