{"id":14312,"date":"2018-04-09T18:17:39","date_gmt":"2018-04-09T17:17:39","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14312"},"modified":"2022-01-11T14:27:46","modified_gmt":"2022-01-11T14:27:46","slug":"um-paralelepipedo-e-uma-piramide","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14312","title":{"rendered":"Um paralelep\u00edpedo e uma pir\u00e2mide"},"content":{"rendered":"<p><ul id='GTTabs_ul_14312' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14312' class='GTTabs_curr'><a  id=\"14312_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14312' ><a  id=\"14312_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_14312'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14314\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14314\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png\" data-orig-size=\"462,355\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Paralelep\u00edpedo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png\" class=\"alignright size-medium wp-image-14314\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6-300x231.png\" alt=\"\" width=\"300\" height=\"231\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6-300x231.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png 462w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura, est\u00e3o representados um paralelep\u00edpedo [<em>ABCDEFGH<\/em>] e uma pir\u00e2mide [<em>HDPC<\/em>], sendo <em>P<\/em> um ponto de [<em>AB<\/em>].<\/p>\n<ol>\n<li>Qual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira?<br \/>\n[A] As retas <em>DP<\/em> e <em>BC<\/em> s\u00e3o concorrentes.<br \/>\n[B] As retas <em>DP<\/em> e <em>BC<\/em> s\u00e3o n\u00e3o complanares.<br \/>\n[C] As retas <em>AB<\/em> e <em>HG<\/em> s\u00e3o concorrentes.<br \/>\n[D] As retas <em>AB<\/em> e <em>HG<\/em> s\u00e3o n\u00e3o complanares.<\/li>\n<li>Admite que :<br \/>\n&#8211;\u00a0\\(\\overline {DP} = 5\\) cm;<br \/>\n&#8211;\u00a0\\(D\\widehat PH = 32^\\circ \\).<br \/>\nDetermina a \u00e1rea do tri\u00e2ngulo [<em>DPH<\/em>].<br \/>\nApresenta o resultado em cm<sup>2<\/sup>, arredondado \u00e0s d\u00e9cimas.<br \/>\nApresenta os c\u00e1lculos que efetuares.<\/li>\n<li>Admite que o volume da pir\u00e2mide [<em>HDPC<\/em>] \u00e9 igual a 10 cm<sup>3<\/sup>.<br \/>\nQual \u00e9 o volume, em cm<sup>3<\/sup>, do paralelep\u00edpedo [<em>ABCDEFGH<\/em>]?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14312' onClick='GTTabs_show(1,14312)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14312'>\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\/2018\/04\/9V2Pag68-6.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14314\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14314\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png\" data-orig-size=\"462,355\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Paralelep\u00edpedo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png\" class=\"alignright size-medium wp-image-14314\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6-300x231.png\" alt=\"\" width=\"300\" height=\"231\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6-300x231.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6.png 462w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>A afirma\u00e7\u00e3o verdadeira \u00e9 a [A].<br \/>\n[A] As retas <em>DP<\/em> e <em>BC<\/em> s\u00e3o concorrentes.<br \/>\n\u00ad<\/li>\n<li>No tri\u00e2ngulo ret\u00e2ngulo [GHP], temos:<br \/>\n\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm tg}\\nolimits} D\\widehat PH = \\frac{{\\overline {DH} }}{{\\overline {DP} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm tg}\\nolimits} 32^\\circ = \\frac{{\\overline {DH} }}{5}}\\\\{}&amp; \\Leftrightarrow &amp;{\\overline {DH} = 5 \\times {\\mathop{\\rm tg}\\nolimits} 32^\\circ }\\end{array}\\]<br \/>\nPortanto, a \u00e1rea do tri\u00e2ngulo [<em>DPH<\/em>],\u00a0em cm<sup>2<\/sup>, arredondado \u00e0s d\u00e9cimas, \u00e9:<br \/>\n\\[{A_{\\left[ {DPH} \\right]}} = \\frac{{\\overline {DP} \\times \\overline {DH} }}{2} = \\frac{{5 \\times 5 \\times {\\mathop{\\rm tg}\\nolimits} 32^\\circ }}{2} \\approx 7,8\\]<\/li>\n<li>\\({V_{\\left[ {ABCDEFGH} \\right]}} = 60\\) cm<sup>3<\/sup>, pois:<br \/>\n\\[\\frac{{{V_{\\left[ {HDPC} \\right]}}}}{{{V_{\\left[ {ABCDEFGH} \\right]}}}} = \\frac{{\\frac{1}{3} \\times {A_{\\left[ {CDP} \\right]}} \\times \\overline {DH} }}{{{A_{\\left[ {ABCD} \\right]}} \\times \\overline {DH} }} = \\frac{{\\frac{1}{3} \\times \\frac{1}{2} \\times A\\left[ {_{ABCD}} \\right] \\times \\overline {DH} }}{{{A_{\\left[ {ABCD} \\right]}} \\times \\overline {DH} }} = \\frac{1}{6}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14312' onClick='GTTabs_show(0,14312)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura, est\u00e3o representados um paralelep\u00edpedo [ABCDEFGH] e uma pir\u00e2mide [HDPC], sendo P um ponto de [AB]. Qual das afirma\u00e7\u00f5es seguintes \u00e9 verdadeira? [A] As retas DP e BC s\u00e3o concorrentes.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14315,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,491,493,490,492,109],"series":[],"class_list":["post-14312","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-cosseno","tag-razao-trigonometrica","tag-seno","tag-tangente","tag-volume"],"views":2961,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag68-6a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14312","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=14312"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14312\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14315"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14312"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14312"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14312"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}