{"id":11861,"date":"2014-04-30T21:37:58","date_gmt":"2014-04-30T20:37:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11861"},"modified":"2022-01-22T16:35:27","modified_gmt":"2022-01-22T16:35:27","slug":"um-triangulo-inscrito-numa-semicircunferencia","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11861","title":{"rendered":"Um tri\u00e2ngulo inscrito numa semicircunfer\u00eancia"},"content":{"rendered":"<p><ul id='GTTabs_ul_11861' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11861' class='GTTabs_curr'><a  id=\"11861_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11861' ><a  id=\"11861_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_11861'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11862\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11862\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\" data-orig-size=\"314,277\" 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=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\" class=\"alignright wp-image-11862\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\" alt=\"Tri\u00e2ngulo\" width=\"240\" height=\"212\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png 314w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16-300x264.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16-150x132.png 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Considere o tri\u00e2ngulo da figura inscrito numa semicircunfer\u00eancia de centro C.<\/p>\n<ol>\n<li>Justifique que o tri\u00e2ngulo \u00e9 ret\u00e2ngulo.<\/li>\n<li>Exprima a \u00e1rea do tri\u00e2ngulo em fun\u00e7\u00e3o do raio e do cateto de comprimento $x$.<\/li>\n<li>Qual deve ser o raio da circunfer\u00eancia para que o tri\u00e2ngulo tenha \u00e1rea $10$ e um cateto seja duplo do outro?<\/li>\n<li>Se o raio for igual a $5$, qual \u00e9 a maior \u00e1rea do tri\u00e2ngulo inscrito?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11861' onClick='GTTabs_show(1,11861)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11861'>\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\/2014\/04\/11-2-pag174-16.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11862\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11862\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\" data-orig-size=\"314,277\" 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=\"Tri\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\" class=\"alignright wp-image-11862\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png\" alt=\"Tri\u00e2ngulo\" width=\"240\" height=\"212\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16.png 314w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16-300x264.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11-2-pag174-16-150x132.png 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>O \u00e2ngulo RPQ \u00e9 reto, pois est\u00e1 inscrito num arco de semicircunfer\u00eancia. Logo, o tri\u00e2ngulo [RPQ] \u00e9 ret\u00e2ngulo.<br \/>\n\u00ad<\/li>\n<li>Como $\\overline {PR}\u00a0 = \\sqrt {{{\\left( {2r} \\right)}^2} &#8211; {x^2}}\u00a0 = \\sqrt {4{r^2} &#8211; {x^2}} $, ent\u00e3o a \u00e1rea do tri\u00e2ngulo pode ser expressa por: \\[a\\left( x \\right) = \\frac{{x\\sqrt {4{r^2} &#8211; {x^2}} }}{2}\\]<br \/>\n\u00ad<\/li>\n<li>Seja\u00a0$\\overline {PR}\u00a0 = 2\\overline {PQ}\u00a0 = 2x = \\sqrt {4{r^2} &#8211; {x^2}} $ e ${a\\left( x \\right) = 10}$.<br \/>\nAssim, vem: \\[\\begin{array}{*{20}{l}}<br \/>\n{a\\left( x \\right) = 10}&amp; \\Leftrightarrow &amp;{\\frac{{x \\times 2x}}{2} = 10} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\sqrt {10} }<br \/>\n\\end{array}\\]<br \/>\nLogo: \\[r = \\frac{{\\sqrt {{{\\left( {\\sqrt {10} } \\right)}^2} + {{\\left( {2\\sqrt {10} } \\right)}^2}} }}{2} = \\frac{{\\sqrt {10 + 40} }}{2} = \\frac{{5\\sqrt 2 }}{2}\\]<br \/>\n\u00ad<\/li>\n<li>Para $r = 5$, tem-se: \\[a\\left( x \\right) = \\frac{{x\\sqrt {100 &#8211; {x^2}} }}{2}\\]<br \/>\nRecorrendo \u00e0 calculadora gr\u00e1fica,\u00a0obt\u00e9m-se:<\/p>\n<table class=\" aligncenter\">\n<tbody>\n<tr>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Janela1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11863\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11863\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Janela1.jpg\" data-orig-size=\"264,136\" 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=\"Janela\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Janela1.jpg\" class=\"aligncenter wp-image-11863 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Janela1.jpg\" alt=\"Janela\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Janela1.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Janela1-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1a.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11864\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11864\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1a.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1a.jpg\" class=\"aligncenter wp-image-11864 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1a.jpg\" alt=\"Gr\u00e1fico 1\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1a.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1a-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11865\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11865\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1b.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1b.jpg\" class=\"aligncenter wp-image-11865 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1b.jpg\" alt=\"Gr\u00e1fico 2\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1b.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/Graf1b-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Admite-se, portanto, que \u00e9 $25$ u.a. a\u00a0 \u00e1rea m\u00e1xima\u00a0do tri\u00e2ngulo.<\/p>\n<\/li>\n<\/ol>\n<p><strong>Extens\u00e3o<\/strong>:<br \/>\nConfirme\u00a0que a \u00e1rea m\u00e1xima \u00e9, de facto, $25$ e que o maximizante \u00e9 $x = 5\\sqrt 2 $ (situa\u00e7\u00e3o em que o tri\u00e2ngulo \u00e9 ret\u00e2ngulo is\u00f3sceles), considerando que a derivada da fun\u00e7\u00e3o \u00e9 definida por: \\[a&#8217;\\left( x \\right) = \\frac{{50 &#8211; {x^2}}}{{\\sqrt {100 &#8211; {x^2}} }}\\]<\/p>\n<\/p>\n<p 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Justifique que o tri\u00e2ngulo \u00e9 ret\u00e2ngulo. Exprima a \u00e1rea do tri\u00e2ngulo em fun\u00e7\u00e3o do raio e do cateto de comprimento&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20896,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,157],"tags":[422,144,158],"series":[],"class_list":["post-11861","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-com-radicais","tag-11-o-ano","tag-extremos-relativos","tag-funcoes-com-radicais-2"],"views":4632,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/04\/11V2Pag174-16_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11861","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=11861"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11861\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20896"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11861"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11861"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11861"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11861"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}