{"id":6388,"date":"2010-12-14T21:26:53","date_gmt":"2010-12-14T21:26:53","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6388"},"modified":"2022-01-21T23:09:01","modified_gmt":"2022-01-21T23:09:01","slug":"um-prisma-quadrangular-regular-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6388","title":{"rendered":"Um prisma quadrangular regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_6388' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6388' class='GTTabs_curr'><a  id=\"6388_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6388' ><a  id=\"6388_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_6388'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6389\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6389\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg\" data-orig-size=\"354,322\" 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=\"Prisma\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg\" class=\"alignright wp-image-6389\" title=\"Prisma\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56-300x272.jpg\" alt=\"\" width=\"240\" height=\"218\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56-300x272.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56-150x136.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg 354w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Num referencial o.n. do espa\u00e7o s\u00e3o dados os pontos A, B, C e H.<\/p>\n<p>Sejam [ABCD] e [EFGH] as bases de um prisma quadrangular regular.<\/p>\n<ol>\n<li>Indique as coordenadas dos pontos E, F e G.<\/li>\n<li>Determine uma equa\u00e7\u00e3o cartesiana do lugar geom\u00e9trico dos pontos equidistantes de H e de B.<\/li>\n<li>Calcule a \u00e1rea do tri\u00e2ngulo [ADG].<\/li>\n<li>Escreva uma condi\u00e7\u00e3o que defina o lugar geom\u00e9trico dos pontos $P\\,(x,y,z)$ tal que $\\overrightarrow{HP}.\\overrightarrow{BP}=0$ e caracterize-o.<\/li>\n<li>Indique dois pontos da recta HB [que n\u00e3o sejam nem H nem B].<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6388' onClick='GTTabs_show(1,6388)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6388'>\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\/pag187-56.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6389\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6389\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg\" data-orig-size=\"354,322\" 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=\"Prisma\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg\" class=\"alignright wp-image-6389\" title=\"Prisma\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56-300x272.jpg\" alt=\"\" width=\"240\" height=\"218\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56-300x272.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56-150x136.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-56.jpg 354w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>As coordenadas desses pontos s\u00e3o: $E\\,(1,1,1)$, $F\\,(2,2,1)$ e $G\\,(1,3,1)$.<br \/>\n\u00ad<\/li>\n<li>O lugar geom\u00e9trico dos pontos equidistantes de H e de B \u00e9 o plano mediador do segmento [HB].<br \/>\nPortanto, \u00e9 o lugar geom\u00e9trico dos pontos $P\\,(x,y,z)$ tais que $\\overrightarrow{HB}.\\overrightarrow{MP}=0$, sendo $M\\,(1,2,\\frac{1}{2})$ o ponto m\u00e9dio do segmento [HB].<br \/>\nOra,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{HB}.\\overrightarrow{MP}=0 &amp; \\Leftrightarrow\u00a0 &amp; (2,0,-1).(x-1,y-2,z-\\frac{1}{2})=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 2x-2-z+\\frac{1}{2}=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 4x-2z-3=0\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, $4x-2z-3=0$ \u00e9 uma equa\u00e7\u00e3o do plano mediador de [HB].<br \/>\n\u00ad<\/li>\n<li>A \u00e1rea do tri\u00e2ngulo [ADG] \u00e9:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n{{A}_{[ADG]}} &amp; = &amp; \\frac{\\overline{AD}\\times \\overline{DG}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\overline{BC}\\times \\overline{AF}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{{{(1-2)}^{2}}+{{(3-2)}^{2}}+0}\\times \\sqrt{{{(2-1)}^{2}}+{{(2-1)}^{2}}+{{(1-0)}^{2}}}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{2}\\times \\sqrt{3}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{6}}{2}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>O lugar geom\u00e9trico dos pontos $P\\,(x,y,z)$ tal que $\\overrightarrow{HP}.\\overrightarrow{BP}=0$ \u00e9 a superf\u00edcie esf\u00e9rica de di\u00e2metro [HB].<br \/>\nO seu centro \u00e9 $M\\,(1,2,\\frac{1}{2})$ e o raio \u00e9 $r=\\frac{\\left\\| \\overrightarrow{HB} \\right\\|}{2}=\\frac{\\sqrt{{{2}^{2}}+1}}{2}=\\frac{\\sqrt{5}}{2}$, pelo que pode ser definida pela equa\u00e7\u00e3o ${{(x-1)}^{2}}+{{(y-2)}^{2}}+{{(z-\\frac{1}{2})}^{2}}=\\frac{5}{4}$.<br \/>\n\u00ad<br \/>\nEqua\u00e7\u00e3o esta que tamb\u00e9m se pode obter resolvendo a condi\u00e7\u00e3o $\\overrightarrow{HP}.\\overrightarrow{BP}=0$:\u00a0 \\[\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{HP}.\\overrightarrow{BP}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x,y-2,z-1).(x-2,y-2,z)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{x}^{2}}-2x+{{(y-2)}^{2}}+{{z}^{2}}-z=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(x-1)}^{2}}-1+{{(y-2)}^{2}}+{{(z-\\frac{1}{2})}^{2}}-\\frac{1}{4}=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{(x-1)}^{2}}+{{(y-2)}^{2}}+{{(z-\\frac{1}{2})}^{2}}=\\frac{5}{4}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>Por exemplo, os pontos:<br \/>\n$S=B+\\overrightarrow{HB}=(2,2,0)+(2,0,-1)=(4,2,-1)$;<br \/>\n$T=B+2\\times \\overrightarrow{HB}=(2,2,0)+2\\times (2,0,-1)=(6,2,-2)$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6388' onClick='GTTabs_show(0,6388)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num referencial o.n. do espa\u00e7o s\u00e3o dados os pontos A, B, C e H. Sejam [ABCD] e [EFGH] as bases de um prisma quadrangular regular. Indique as coordenadas dos pontos E,&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20842,"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-6388","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":5358,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag187-56_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6388","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=6388"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6388\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20842"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6388"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6388"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6388"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6388"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}