{"id":6382,"date":"2010-12-14T15:39:58","date_gmt":"2010-12-14T15:39:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6382"},"modified":"2022-01-21T22:57:45","modified_gmt":"2022-01-21T22:57:45","slug":"uma-piramide-quadrangular-regular-4","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6382","title":{"rendered":"Uma pir\u00e2mide quadrangular regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_6382' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6382' class='GTTabs_curr'><a  id=\"6382_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6382' ><a  id=\"6382_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_6382'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6383\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6383\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53.jpg\" data-orig-size=\"323,414\" 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\/pag187-53.jpg\" class=\"alignright wp-image-6383\" title=\"Pir\u00e2mide\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53-234x300.jpg\" alt=\"\" width=\"300\" height=\"385\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53-234x300.jpg 234w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53-117x150.jpg 117w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53.jpg 323w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Considere Oxyz um referencial ortonormado.<\/p>\n<p>A pir\u00e2mide quadrangular regular est\u00e1 assente sobre um plano paralelo a xOy, tem o v\u00e9rtice no eixo Oz e os planos xOz e yOz s\u00e3o planos mediadores das arestas da base (como ilustra a figura).<\/p>\n<p>Conhecem-se ainda $A\\,(1,1,3)$ e a altura da pir\u00e2mide, que \u00e9 5 unidades.<\/p>\n<ol>\n<li>Caracterize, por uma condi\u00e7\u00e3o, o plano em que a base da pir\u00e2mide est\u00e1 assente.<\/li>\n<li>Identifique, pelas suas coordenadas, os outros v\u00e9rtices da pir\u00e2mide.<\/li>\n<li>Calcule a medida do comprimento da aresta lateral da pir\u00e2mide.<\/li>\n<li>Calcule o volume da pir\u00e2mide.<\/li>\n<li>Escreva equa\u00e7\u00f5es cartesianas da recta que cont\u00e9m a aresta lateral AV.<\/li>\n<li>Determine um vector normal ao plano $\\alpha $ da face [ABV] e escreva uma equa\u00e7\u00e3o desse plano.<\/li>\n<li>Considere a esfera que tem por c\u00edrculo m\u00e1ximo o c\u00edrculo circunscrito \u00e0 base da pir\u00e2mide e escreva uma condi\u00e7\u00e3o que a defina.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6382' onClick='GTTabs_show(1,6382)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6382'>\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\/2010\/12\/pag187-53.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6383\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6383\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53.jpg\" data-orig-size=\"323,414\" 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\/pag187-53.jpg\" class=\"alignright wp-image-6383\" title=\"Pir\u00e2mide\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53-234x300.jpg\" alt=\"\" width=\"300\" height=\"385\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53-234x300.jpg 234w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53-117x150.jpg 117w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-53.jpg 323w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>O\u00a0plano em que a base da pir\u00e2mide est\u00e1 assente pode ser caracterizado pela condi\u00e7\u00e3o $z=3$.<br \/>\n\u00ad<\/li>\n<li>Os outros v\u00e9rtices da pir\u00e2mide t\u00eam as seguintes coordenadas:\n<p>$B\\,(-1,1,3)$, $C\\,(-1,-1,3)$, $D\\,(1,-1,3)$ e $V\\,(0,0,8)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A medida da aresta a pir\u00e2mide \u00e9:\n<p>$\\overline{VA}=\\sqrt{{{(1-0)}^{2}}+{{(1-0)}^{2}}+{{(3-8)}^{2}}}=\\sqrt{27}=3\\sqrt{3}$ unidades.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O volume da pir\u00e2mide, em unidades c\u00fabicas, \u00e9:<br \/>\n\\[V=\\frac{1}{3}\\times \\overline{AB}\\times \\overline{AD}\\times \\overline{V&#8217;V}=\\frac{1}{3}\\times 2\\times 2\\times 5=\\frac{20}{3}\\]<br \/>\n(V\u00b4\u00e9 a projec\u00e7\u00e3o de V sobre a base da pir\u00e2mide.)<br \/>\n\u00ad<\/li>\n<li>Como $A\\,(1,1,3)$ e $V\\,(0,0,8)$, ent\u00e3o $\\overrightarrow{AV}=(-1,-1,5)$.\n<p>Logo, uma equa\u00e7\u00e3o vetorial da recta AV \u00e9 $(x,y,z)=(1,1,3)+k(-1,-1,5)\\,,\\,\\,k\\in \\mathbb{R}$.<\/p>\n<p>Desta equa\u00e7\u00e3o resulta:<br \/>\n\\[\\begin{matrix}<br \/>\n\\frac{x-1}{-1}=\\frac{y-1}{-1}=\\frac{z-3}{5} &amp; \\Leftrightarrow\u00a0 &amp; -x+1=-y+1=\\frac{z-3}{5} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix}<br \/>\ny=x &amp; \\wedge\u00a0 &amp; 5y+z-8=0\u00a0 \\\\<br \/>\n\\end{matrix}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<br \/>\nLogo, $\\begin{matrix}<br \/>\ny=x &amp; \\wedge\u00a0 &amp; 5y+z-8=0\u00a0 \\\\<br \/>\n\\end{matrix}$ s\u00e3o equa\u00e7\u00f5es cartesianas da recta pedida.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora, $\\overrightarrow{AV}=(-1,-1,5)$ e $\\overrightarrow{AB}=(-2,0,0)$.\n<p>Determinemos um vetor normal ao plano considerado, isto \u00e9, um vetor $\\vec{n}=(a,b,c)$\u00a0 perpendicular aos vetores $\\overrightarrow{AV}=(-1,-1,5)$ e $\\overrightarrow{AB}=(-2,0,0)$.<\/p>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c).(-1,-1,5)=0\u00a0 \\\\<br \/>\n(a,b,c).(-2,0,0)=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n-a-b+5c=0\u00a0 \\\\<br \/>\n-2a=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=0\u00a0 \\\\<br \/>\nb=5c\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, $\\overrightarrow{{{n}_{1}}}=(0,5,1)$, por exemplo, \u00e9 um vetor perpendicular ao plano ABV.<\/p>\n<p>Assim, o plano AVB pode ser caracterizado por uma equa\u00e7\u00e3o da forma $5y+z+d=0$.<br \/>\nComo A pertence a este plano, vem $5\\times 1+3+d=0\\Leftrightarrow d=-8$.<\/p>\n<p>Logo, $5y+z-8=0$ \u00e9 uma equa\u00e7\u00e3o do plano ABV.<\/p>\n<p>(Note que foi este um dos planos\u00a0utilizados para definir a recta AV, na al\u00ednea anterior. Qual foi o outro plano?)<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora, essa esfera tem centro em $V&#8217;\\,(0,0,3)$ e raio $r=\\frac{\\overline{BD}}{2}=\\frac{2\\sqrt{2}}{2}=\\sqrt{2}$.\n<p>Logo, essa esfera pode ser definida pela condi\u00e7\u00e3o ${{x}^{2}}+{{y}^{2}}+{{(z-3)}^{2}}\\le 2$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6382' onClick='GTTabs_show(0,6382)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere Oxyz um referencial ortonormado. A pir\u00e2mide quadrangular regular est\u00e1 assente sobre um plano paralelo a xOy, tem o v\u00e9rtice no eixo Oz e os planos xOz e yOz s\u00e3o planos&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20840,"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-6382","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":5300,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag187-53_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6382","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=6382"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6382\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20840"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6382"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6382"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6382"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6382"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}