{"id":6175,"date":"2010-11-27T01:24:51","date_gmt":"2010-11-27T01:24:51","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6175"},"modified":"2022-01-04T15:22:10","modified_gmt":"2022-01-04T15:22:10","slug":"um-triangulo-rectangulo-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6175","title":{"rendered":"Um tri\u00e2ngulo ret\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_6175' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6175' class='GTTabs_curr'><a  id=\"6175_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6175' ><a  id=\"6175_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_6175'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>As medidas dos catetos de um tri\u00e2ngulo ret\u00e2ngulo s\u00e3o $\\overline{AB}=3,6\\,m$ e $\\overline{BC}=4,8\\,m$.<\/p>\n<p>Calcula:<\/p>\n<ol>\n<li>a medida da hipotenusa [AC];<\/li>\n<li>a medida da altura [BH] relativa \u00e0 hipotenusa;<\/li>\n<li>as medidas dos segmentos [AH] e [HC].<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6175' onClick='GTTabs_show(1,6175)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6175'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><span class=\"alignright\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"text-align: right\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/span>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [ABC], temos:\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n{{\\overline{AC}}^{2}}={{3,6}^{2}}+{{4,8}^{2}} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AC}=\\sqrt{{{3,6}^{2}}+{{4,8}^{2}}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AC}=\\sqrt{36}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AC}=6\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Logo, $\\overline{AC}=6\\,m$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A altura relativa \u00e0 hipotenusa divide o tri\u00e2ngulo ret\u00e2ngulo em dois tri\u00e2ngulos ret\u00e2ngulos semelhantes entre si e semelhantes ao tri\u00e2ngulo ret\u00e2ngulo inicial.<br \/>\nConsiderando os tri\u00e2ngulos [ABH] e [ABC], tem-se: $\\frac{\\overline{AB}}{\\overline{AC}}=\\frac{\\overline{HB}}{\\overline{CB}}$.<\/p>\n<p>Logo:<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\frac{3,6}{6}=\\frac{\\overline{HB}}{4,8} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{HB}=\\frac{3,6\\times 4,8}{6}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{HB}=2,88\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, $\\overline{HB}=2,88\\,m$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [ABH], temos: ${{\\overline{AH}}^{2}}+{{\\overline{HB}}^{2}}={{\\overline{AB}}^{2}}$.<br \/>\nLogo, \\[\\begin{array}{*{35}{l}}<br \/>\n{{\\overline{AH}}^{2}}+{{2,88}^{2}}={{3,6}^{2}} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AH}=\\sqrt{{{3,6}^{2}}-{{2,88}^{2}}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AH}=\\sqrt{4,6656}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AH}=2,16\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto,\u00a0$\\overline{AH}=2,16\\,m$.<\/p>\n<p>Logo, $\\overline{HC}=\\overline{AC}-\\overline{AH}=6-2,16=3,84\\,m$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6175' onClick='GTTabs_show(0,6175)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado As medidas dos catetos de um tri\u00e2ngulo ret\u00e2ngulo s\u00e3o $\\overline{AB}=3,6\\,m$ e $\\overline{BC}=4,8\\,m$. Calcula: a medida da hipotenusa [AC]; a medida da altura [BH] relativa \u00e0 hipotenusa; as medidas dos segmentos [AH]&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14114,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,112],"tags":[424,67,118],"series":[],"class_list":["post-6175","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-decomposicao-de-figuras-teorema-de-pitagoras","tag-8-o-ano","tag-geometria","tag-teorema-de-pitagoras"],"views":1863,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat56.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6175","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=6175"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6175\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6175"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}