{"id":2250,"date":"2020-04-20T11:07:34","date_gmt":"2020-04-20T09:07:34","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2250"},"modified":"2021-03-02T16:45:52","modified_gmt":"2021-03-02T15:45:52","slug":"trouver-une-aire-2","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/04\/20\/trouver-une-aire-2\/","title":{"rendered":"Trouver une aire"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Comment exprimer en fonction de <em>x<\/em>, <em>y<\/em> et <em>z<\/em> l&rsquo;aire <em>w<\/em> ?<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle-1024x641.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"641\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle-1024x641.png\" alt=\"\" class=\"wp-image-2251\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle-1024x641.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle-300x188.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle-600x376.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle-768x481.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle.png 1110w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ce probl\u00e8me peut \u00eatre abord\u00e9 comme application dans le chapitre des polyn\u00f4mes de degr\u00e9 2&#8230;<\/p>\n\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\/2020\/04\/20\/trouver-une-aire-2\/#Etape_1\" >\u00c9tape 1<\/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\/04\/20\/trouver-une-aire-2\/#Etape_2\" >\u00c9tape 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/04\/20\/trouver-une-aire-2\/#Etape_3\" >\u00c9tape 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/04\/20\/trouver-une-aire-2\/#Etape_4\" >\u00c9tape 4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/04\/20\/trouver-une-aire-2\/#Code_LaTeX_de_la_figure\" >Code \\(\\LaTeX\\) de la figure<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_1\"><\/span>\u00c9tape 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Regardons d&rsquo;abord le triangle bleu d&rsquo;aire <em>x<\/em> : c&rsquo;est un triangle rectangle de c\u00f4t\u00e9s perpendiculaires de mesure <em>a<\/em> et <em>t<\/em>, tels que:$$\\frac{1}{2}at=x,$$donc:$$t=\\frac{2x}{a}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_2\"><\/span>\u00c9tape 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Regardons \u00e0 pr\u00e9sent le triangle vert, d&rsquo;aire <em>y<\/em> : il est aussi rectangle et le c\u00f4t\u00e9 de droite vaut (on utilise le m\u00eame raisonnement qu&rsquo;\u00e0 l&rsquo;\u00e9tape 1) :$$\\frac{2y}{b}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_3\"><\/span>\u00c9tape 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On d\u00e9duit maintenant les dimensions restantes:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2-1024x639.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"639\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2-1024x639.png\" alt=\"\" class=\"wp-image-2252\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2-1024x639.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2-300x187.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2-600x374.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2-768x479.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/04\/rectangle2.png 1162w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, l&rsquo;aire du triangle rouge valant <em>z<\/em>, on a l&rsquo;\u00e9galit\u00e9:$$\\frac{1}{2}\\left(b-\\frac{2x}{a}\\right)\\left(a-\\frac{2y}{b}\\right)=z$$que l&rsquo;on peut aussi \u00e9crire, en multipliant par 2 les deux membres de l&rsquo;\u00e9galit\u00e9:$$\\left(b-\\frac{2x}{a}\\right)\\left(a-\\frac{2y}{b}\\right)=2z.$$En d\u00e9veloppant le membre de gauche, on arrive \u00e0:$$ab-2y-2x+\\frac{4xy}{ab}=2z$$ou encore:$$ab-2(x+y+z)+\\frac{4xy}{ab}=0.$$En multipliant maintenant par <em>ab<\/em> les deux membres de cette derni\u00e8re \u00e9galit\u00e9, on obtient:$$(ab)^2-2(x+y+z)(ab)+4xy=0.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_4\"><\/span>\u00c9tape 4<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Posons <em>T<\/em> = <em>ab<\/em> dans cette derni\u00e8re \u00e9galit\u00e9. Alors, on a:$$T^2-2(x+y+z)T+4xy=0.$$C&rsquo;est une \u00e9quation quadratique d&rsquo;inconnue <em>T<\/em> dont les solutions peuvent \u00eatre:$$T_{1,2}=\\frac{2(x+y+z)\\pm\\sqrt{4(x+y+z)^2-16xy}}{2}$$ soit:$$T_{1,2}=(x+y+z)\\pm\\sqrt{(x+y+z)^2-4xy}.$$ Ne perdons pas de vue que <em>T<\/em> = <em>ab<\/em> = <em>x+y+z+w<\/em> et donc que<em>:<\/em>$$T>x+y+z.$$Ainsi, la seule solution possible est:$$T=(x+y+z)+\\sqrt{(x+y+z)^2-4xy}.$$On en d\u00e9duit alors que:$$\\boxed{w=\\sqrt{(x+y+z)^2-4xy}}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Code_LaTeX_de_la_figure\"><\/span>Code \\(\\LaTeX\\) de la figure<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{standalone}\n\\usepackage{pas-math}\n\\usepackage{tikz}\n\\begin{document}\n\\begin{tikzpicture}\n\\clip (-0.5,-1) rectangle (11.5,6.5);\n\\draw (0,0) rectangle +(10,6);\n\\node[left] at (0,3) {$a$};\n\\node[above] at (5,6) {$b$};\n\\fill[blue!20] (0,0) -- (0,6) -- (4,0) -- cycle;\n\\fill[red!20] (4,0) -- (10,0) -- (10,4.5) -- cycle;\n\\fill[green!20] (0,6) -- (10,6) -- (10,4.5) -- cycle;\n\\draw (0,0) rectangle +(10,6);\n\\node at (1.5,2) {$x$};\n\\node at (8,1.5) {$z$};\n\\node at (7,5.5) {$y$};\n\\node at (5,3) {$w$};\n\\draw[&amp;lt;-&amp;gt;,&amp;gt;=latex] (0,-0.5) -- (4,-0.5) node[midway,fill=white] {$\\dfrac{2x}{a}$};\n\\draw[&amp;lt;-&amp;gt;,&amp;gt;=latex] (4,-0.5) -- (10,-0.5) node[midway,fill=white] {$b-\\dfrac{2x}{a}$};\n\\draw[&amp;lt;-&amp;gt;,&amp;gt;=latex] (10.75,4.5) -- (10.75,6) node[midway,fill=white] {$\\dfrac{2y}{b}$};\n\\draw[&amp;lt;-&amp;gt;,&amp;gt;=latex] (10.75,4.5) -- (10.75,0) node[midway,fill=white] {$a-\\dfrac{2y}{b}$};\n\\end{tikzpicture}\n\\end{document}<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Comment exprimer en fonction de x, y et z l&rsquo;aire w ? Ce probl\u00e8me peut \u00eatre abord\u00e9 comme application dans le chapitre des polyn\u00f4mes de degr\u00e9 2&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[54,80,203,40,46,157],"class_list":["post-2250","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-aire","tag-polynome","tag-rectangle","tag-second-degre","tag-triangle","tag-triangle-rectangle"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trouver une aire - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/04\/20\/trouver-une-aire-2\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trouver une aire - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Comment exprimer en fonction de x, y et z l&rsquo;aire w ? 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