{"id":6364,"date":"2010-12-12T17:06:39","date_gmt":"2010-12-12T17:06:39","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6364"},"modified":"2022-01-21T18:49:09","modified_gmt":"2022-01-21T18:49:09","slug":"uma-piramide-quadrangular-regular-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6364","title":{"rendered":"Uma pir\u00e2mide quadrangular regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_6364' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6364' class='GTTabs_curr'><a  id=\"6364_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6364' ><a  id=\"6364_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_6364'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6365\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6365\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg\" data-orig-size=\"328,395\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Pir\u00e2mide\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg\" class=\"alignright wp-image-6365\" title=\"Pir\u00e2mide\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42-249x300.jpg\" alt=\"\" width=\"240\" height=\"289\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42-249x300.jpg 249w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42-124x150.jpg 124w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg 328w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura est\u00e1 representada, em referencial o.n. Oxyz, uma pir\u00e2mide quadrangular regular.<\/p>\n<p>A base da pir\u00e2mide est\u00e1 contida no plano de equa\u00e7\u00e3o $z=4$.<\/p>\n<ul>\n<li>O v\u00e9rtice A pertence ao eixo Oz.<\/li>\n<li>O v\u00e9rtice B pertence ao plano yOz.<\/li>\n<li>O v\u00e9rtice D pertence ao plano xOz.<\/li>\n<li>O v\u00e9rtice C tem coordenadas $(4,4,4)$.<\/li>\n<li>A altura da pir\u00e2mide \u00e9 6.<\/li>\n<\/ul>\n<ol>\n<li>Mostre que uma condi\u00e7\u00e3o que define a reta DE \u00e9 $x-4=-y=\\frac{z-4}{3}$.<\/li>\n<li>Determine uma equa\u00e7\u00e3o do plano que passa no ponto B e \u00e9 perpendicular \u00e0 reta DE.<\/li>\n<li>Determine a \u00e1rea da sec\u00e7\u00e3o produzida na pir\u00e2mide pelo plano xOy.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6364' onClick='GTTabs_show(1,6364)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6364'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Como $D\\,(4,0,4)$ e $E\\,(2,2,-2)$, ent\u00e3o o vetor $\\overrightarrow{DE}=(-2,2,-6)$ \u00e9 diretor da reta DE.\n<p>Uma equa\u00e7\u00e3o vetorial da reta DE \u00e9 $(x,y,z)=(4,0,4)+k(-2,2,-6)\\,,\\,\\,k\\in \\mathbb{R}$, donde se obt\u00e9m:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=4-2k\u00a0 \\\\<br \/>\ny=0+2k\u00a0 \\\\<br \/>\nz=4-6k\u00a0 \\\\<br \/>\n\\end{array}\\,,\\,\\,k\\in \\mathbb{R} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nk=\\frac{x-4}{-2}\u00a0 \\\\<br \/>\nk=\\frac{y}{2}\u00a0 \\\\<br \/>\nk=\\frac{z-4}{-6}\u00a0 \\\\<br \/>\n\\end{array}\\,,\\,\\,k\\in \\mathbb{R} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\frac{x-4}{-2}=\\frac{y}{2}=\\frac{z-4}{-6} &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{x-4}{-1}=\\frac{y}{1}=\\frac{z-4}{-3} &amp; \\Leftrightarrow\u00a0 &amp; x-4=-y=\\frac{z-4}{3} &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto, $x-4=-y=\\frac{z-4}{3}$ \u00e9 uma condi\u00e7\u00e3o que define a reta DE.<\/p>\n<p><strong><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6365\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6365\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg\" data-orig-size=\"328,395\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Pir\u00e2mide\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg\" class=\"alignright wp-image-6365\" title=\"Pir\u00e2mide\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42-249x300.jpg\" alt=\"\" width=\"240\" height=\"289\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42-249x300.jpg 249w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42-124x150.jpg 124w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag184-42.jpg 328w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Alternativa<\/strong>:<br \/>\nBasta mostrar que as coordenadas dos pontos D e E verificam a condi\u00e7\u00e3o apresentada:<br \/>\n$4-4=-0=\\frac{4-4}{3}$ \u00e9 uma proposi\u00e7\u00e3o verdadeira;<br \/>\n$2-4=-2=\\frac{-2-4}{3}$ \u00e9 tamb\u00e9m uma proposi\u00e7\u00e3o verdadeira.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Um vetor normal ao plano pedido \u00e9 $\\overrightarrow{DE}=(-2,2,-6)$.<br \/>\nUm ponto do plano \u00e9 $B\\,(0,4,4)$.<\/p>\n<p>Designando $P\\,(x,y,z)$ um ponto gen\u00e9rico do plano pedido, ser\u00e1 $\\overrightarrow{BP}.\\overrightarrow{DE}=0$, pois $\\overrightarrow{BP}\\bot \\overrightarrow{DE}$.<\/p>\n<p>Assim, vem: \\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{BP}.\\overrightarrow{DE}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x,y-4,z-4).(-2,2,-6)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; -2x+2y-8-6z+24=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x-y+3z-8=0\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto, $x-y+3z-8=0$ \u00e9 uma equa\u00e7\u00e3o do plano pedido.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A sec\u00e7\u00e3o produzida na pir\u00e2mide pelo plano xOy \u00e9 um quadrado semelhante \u00e0 base da pir\u00e2mide, com raz\u00e3o de semelhan\u00e7a $r=\\frac{2}{6}=\\frac{1}{3}$.<br \/>\nPortanto, a \u00e1rea da sec\u00e7\u00e3o \u00e9: \\[A={{A}_{[ABCD]}}\\times {{(\\frac{1}{3})}^{2}}=16\\times \\frac{1}{9}=\\frac{16}{9}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6364' onClick='GTTabs_show(0,6364)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representada, em referencial o.n. Oxyz, uma pir\u00e2mide quadrangular regular. A base da pir\u00e2mide est\u00e1 contida no plano de equa\u00e7\u00e3o $z=4$. O v\u00e9rtice A pertence ao eixo Oz. O&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20831,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67],"series":[],"class_list":["post-6364","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria"],"views":4441,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag184-42_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6364","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=6364"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6364\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20831"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6364"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6364"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6364"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}