{"id":10163,"date":"2012-10-06T02:07:15","date_gmt":"2012-10-06T01:07:15","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10163"},"modified":"2022-01-20T22:43:09","modified_gmt":"2022-01-20T22:43:09","slug":"em-volta-de-um-retangulo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10163","title":{"rendered":"Em volta de um ret\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_10163' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10163' class='GTTabs_curr'><a  id=\"10163_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10163' ><a  id=\"10163_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_10163'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10177\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10177\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\" data-orig-size=\"440,295\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349657615&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\" class=\"alignright  wp-image-10177\" title=\"Ret\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\" alt=\"\" width=\"264\" height=\"177\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg 440w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-300x201.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-150x100.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-400x268.jpg 400w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a>Observe a figura ao lado, onde [ABCD] \u00e9 um ret\u00e2ngulo.<\/p>\n<ol>\n<li>Se o segmento de reta [AM] \u00e9 perpendicular a BD, demonstre que os tri\u00e2ngulos [MAD] e [ABD] s\u00e3o semelhantes.<\/li>\n<li>Se AM e PC s\u00e3o paralelas e AM e BD s\u00e3o perpendiculares, demostre que os tri\u00e2ngulos [MAD] e [PBC] s\u00e3o semelhantes e conclua que $\\frac{{\\overline {AD} }}{{\\overline {PC} }} = \\frac{{\\overline {MD} }}{{\\overline {BP} }}$.<\/li>\n<li>Se PC \u00e9 perpendicular a BD, demonstre que $\\frac{{\\overline {NC} }}{{\\overline {NB} }} = \\frac{{\\overline {NB} }}{{\\overline {NP} }}$.<\/li>\n<li>Se AM \u00e9 perpendicular a BD, PC paralela a AM, $\\overline {AM}\u00a0 = 6$ cm e\u00a0$\\overline {BN}\u00a0 = 4$ cm, determine as raz\u00f5es entre os per\u00edmetros e as \u00e1reas dos tri\u00e2ngulos\u00a0[BAD] e [PBC], depois de justificar que os tri\u00e2ngulos s\u00e3o semelhantes.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10163' onClick='GTTabs_show(1,10163)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10163'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10177\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10177\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\" data-orig-size=\"440,295\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349657615&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\" class=\"alignright  wp-image-10177\" title=\"Ret\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg\" alt=\"\" width=\"264\" height=\"177\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.jpg 440w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-300x201.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-150x100.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-400x268.jpg 400w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a>Os dois tri\u00e2ngulos, [MAD] e [ABD], s\u00e3o ambos ret\u00e2ngulos.\n<p>Ora, o \u00e2ngulo ADB \u00e9 comum aos dois tri\u00e2ngulos.<\/p>\n<p>Logo, os tri\u00e2ngulos [MAD] e [ABD] s\u00e3o semelhantes, pois possuem dois \u00e2ngulos respetivamente iguais, cada um a cada um.<\/p>\n<\/li>\n<li>Os dois tri\u00e2ngulos, [MAD] e [PBC], s\u00e3o ambos ret\u00e2ngulos.\n<p>Os \u00e2ngulos DAM e\u00a0BCP s\u00e3o geometricamente iguais, pois s\u00e3o \u00e2ngulos agudos de lados paralelos.<\/p>\n<p>Logo, os tri\u00e2ngulos [MAD] e [PBC] s\u00e3o semelhantes, pois possuem dois \u00e2ngulos respetivamente iguais.<\/p>\n<p>Como em tri\u00e2ngulos semelhantes, os lados correspondentes t\u00eam comprimentos diretamente proporcionais, resulta: $\\frac{{\\overline {AD} }}{{\\overline {PC} }} = \\frac{{\\overline {MD} }}{{\\overline {BP} }} = \\frac{{\\overline {AM} }}{{\\overline {BC} }}$.<\/p>\n<\/li>\n<li>Os\u00a0 tri\u00e2ngulos [PNB] e [BNC] s\u00e3o semelhantes, pois possuem dois \u00e2ngulos respetivamente iguais (os \u00e2ngulos PNB e BNC s\u00e3o retos e os \u00e2ngulos ABD e BCP s\u00e3o tamb\u00e9m geometricamente iguais [por 1. e 2.]).\n<p>Consequentemente, os lados correspondentes desses tri\u00e2ngulos possuem comprimentos diretamente proporcionais: $\\frac{{\\overline {NC} }}{{\\overline {NB} }} = \\frac{{\\overline {NB} }}{{\\overline {NP} }} = \\frac{{\\overline {BC} }}{{\\overline {BP} }}$.<\/p>\n<\/li>\n<li>Os tri\u00e2ngulos [BAD] e [PBC] s\u00e3o semelhantes, pois s\u00e3o ambos semelhantes ao tri\u00e2ngulo [MAD] (por 1. e 2.).\n<p>Os segmentos de reta [AM] e [BN] s\u00e3o, respetivamente, as alturas relativas \u00e0s hipotenusas dos tri\u00e2ngulos semelhantes [BAD] e [PBC].<\/p>\n<p>Ora, a raz\u00e3o entre os seus comprimentos \u00e9 $r = \\frac{{\\overline {AM} }}{{\\overline {BN} }} = \\frac{6}{4} = \\frac{3}{2}$.<\/p>\n<p>Logo, o tri\u00e2ngulo [BAD] \u00e9 uma amplia\u00e7\u00e3o do tri\u00e2ngulo [PBC] com raz\u00e3o $r = \\frac{3}{2}$.<\/p>\n<p>Consequentemente, a raz\u00e3o entre os per\u00edmetros desses tri\u00e2ngulos \u00e9 $r = \\frac{3}{2}$ e a raz\u00e3o entre as suas \u00e1reas \u00e9 ${r^2} = {\\left( {\\frac{3}{2}} \\right)^2} = \\frac{9}{4}$.<\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10200\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10200\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.png\" data-orig-size=\"775,533\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.png\" class=\"aligncenter  wp-image-10200\" title=\"Ret\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.png\" alt=\"\" width=\"465\" height=\"320\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo.png 775w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-300x206.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-150x103.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/voltaretangulo-400x275.png 400w\" sizes=\"auto, (max-width: 465px) 100vw, 465px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10163' onClick='GTTabs_show(0,10163)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Observe a figura ao lado, onde [ABCD] \u00e9 um ret\u00e2ngulo. Se o segmento de reta [AM] \u00e9 perpendicular a BD, demonstre que os tri\u00e2ngulos [MAD] e [ABD] s\u00e3o semelhantes. Se AM&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20787,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,67,430,149],"series":[],"class_list":["post-10163","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-geometria","tag-modulo-inicial","tag-semelhanca-de-triangulos"],"views":1990,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag038-29_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10163","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10163"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10163\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20787"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10163"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}