{"id":7799,"date":"2022-07-09T16:14:39","date_gmt":"2022-07-09T14:14:39","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=7799"},"modified":"2022-07-09T16:15:28","modified_gmt":"2022-07-09T14:15:28","slug":"trouver-la-distance-entre-deux-centres-de-cercles","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/07\/09\/trouver-la-distance-entre-deux-centres-de-cercles\/","title":{"rendered":"Trouver la distance entre deux centres de cercles"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Comment trouver la distance entre les centres des deux cercles ci-dessous ? ici, ces cercles sont tangents \u00e0 la diagonale trac\u00e9e ainsi qu&rsquo;aux c\u00f4t\u00e9s du rectangle.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"645\" height=\"510\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-5.png\" alt=\"distance centres cercles\" class=\"wp-image-7800\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-5.png 645w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-5-300x237.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-5-600x474.png 600w\" sizes=\"auto, (max-width: 645px) 100vw, 645px\" \/><\/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\/2022\/07\/09\/trouver-la-distance-entre-deux-centres-de-cercles\/#Distance_entre_les_centres_des_deux_cercles_calcul_du_rayon\" >Distance entre les centres des deux cercles: calcul du rayon<\/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\/2022\/07\/09\/trouver-la-distance-entre-deux-centres-de-cercles\/#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=\"Distance_entre_les_centres_des_deux_cercles_calcul_du_rayon\"><\/span>Distance entre les centres des deux cercles: calcul du rayon<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Dans un premier temps, je vais calculer le rayon des deux cercles. Pour cela, je vais d\u00e9couper fa\u00e7on puzzle le demi-rectangle au-dessus de la diagonale en trois morceaux comme ci-dessous:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"649\" height=\"513\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-6.png\" alt=\"\" class=\"wp-image-7801\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-6.png 649w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-6-300x237.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-6-600x474.png 600w\" sizes=\"auto, (max-width: 649px) 100vw, 649px\" \/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\"><li>L&rsquo;aire du triangle bleu est: $$ \\frac{\\text{diagonale} \\times r}{2} = \\frac{10r}{2}=5r.$$<\/li><li>De m\u00eame, celle du triangle vert est: $$\\frac{6r}{2}=3r$$et celle du triangle rouge est:$$\\frac{8r}{2}=4r.$$<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, l&rsquo;aire du demi-rectangle est la somme de ces trois aires, soit \\(12r\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or, cette aire vaut \\(\\frac{8\\times6}{2}=24\\). On en d\u00e9duit alors que:$$12r=24\\quad\\text{soit}\\quad r=2.$$<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"642\" height=\"504\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-7.png\" alt=\"\" class=\"wp-image-7804\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-7.png 642w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-7-300x236.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/07\/image-7-600x471.png 600w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors que:$$XY=\\sqrt{2^2+4^2}=\\sqrt{20}=2\\sqrt5.$$<\/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{article}\n\\usepackage{tikz}\n\\begin{document}\n\n\\begin{tikzpicture}\n\\fill[blue!10] (0,0) -- (2,4) -- (8,6) -- cycle;\n\\fill[green!10] (0,0) -- (2,4) -- (0,6) -- cycle;\n\\fill[red!10] (0,6) -- (2,4) -- (8,6) -- cycle;\n%\n\\draw (0,0) rectangle (8,6);\n\\draw (0,0) -- (8,6);\n\\draw (6,2) circle (2cm);\n\\draw (2,4) circle (2cm);\n\\fill (2,4) circle (1pt) node[above right] {$X$};\n\\fill (6,2) circle (1pt) node[above] {$Y$};\n\\draw[thick,red] (2,4) -- (6,2) node[midway,above] {?};\n\\draw[&lt;->,>=latex] (8.5,0) -- (8.5,6) node[midway,right] {6};\n\\draw[&lt;->,>=latex] (0,-.5) -- (8,-.5) node[midway,below] {8};\n%\n\\draw (2,4) -- (3.2,2.4) node[midway,below left] {$r$};\n\\draw[shift={(3.2,2.4)},rotate=37] (.2,0) -- (.2,.2) -- (0,.2);\n%\n\\draw (2,4) -- (0,4) node[midway,above] {$r$};\n\\draw (0,3.8) -- (.2,3.8) -- (.2,4);\n%\n\\draw (2,4) -- (2,6) node[midway,right] {$r$};\n\\draw (2,5.8) -- (2.2,5.8) -- (2.2,6);\n%\n\\draw[very thick] (2,4) -- (2,2) node[midway,fill=white] {2} -- (6,2) node[midway,fill=white] {4};\n\\end{tikzpicture}\n\n\\end{document}<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Comment trouver la distance entre les centres des deux cercles ci-dessous ? ici, ces cercles sont tangents \u00e0 la diagonale trac\u00e9e ainsi qu&rsquo;aux c\u00f4t\u00e9s du rectangle.<\/p>\n","protected":false},"author":1,"featured_media":7807,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[54,60,52,88,156,46],"class_list":["post-7799","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-aire","tag-cercle","tag-geometrie","tag-latex","tag-pythagore","tag-triangle"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trouver la distance entre deux centres de cercles - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Comment trouver la distance entre les centres des deux cercles ci-dessous ? 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