{"id":6126,"date":"2010-11-26T02:50:11","date_gmt":"2010-11-26T02:50:11","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6126"},"modified":"2022-01-19T18:56:29","modified_gmt":"2022-01-19T18:56:29","slug":"rascunho-35","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6126","title":{"rendered":"Determina o per\u00edmetro e a \u00e1rea dos trap\u00e9zios"},"content":{"rendered":"<p><ul id='GTTabs_ul_6126' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6126' class='GTTabs_curr'><a  id=\"6126_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6126' ><a  id=\"6126_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_6126'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Determina o per\u00edmetro e a \u00e1rea de cada um dos seguintes trap\u00e9zios (as medidas est\u00e3o em cent\u00edmetros):<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6129\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6129\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8.jpg\" data-orig-size=\"999,228\" 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=\"Trap\u00e9zios\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8.jpg\" class=\"aligncenter wp-image-6129\" title=\"Trap\u00e9zios\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8.jpg\" alt=\"\" width=\"720\" height=\"164\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8.jpg 999w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8-300x68.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8-150x34.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8-400x91.jpg 400w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><\/a><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6126' onClick='GTTabs_show(1,6126)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6126'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p style=\"text-align: left;\"><strong><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12469\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12469\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8a.png\" data-orig-size=\"985,412\" 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=\"Al\u00ednea a)\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8a.png\" class=\"alignright wp-image-12469 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8a-300x125.png\" alt=\"Al\u00ednea a)\" width=\"300\" height=\"125\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8a-300x125.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8a.png 985w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>a)<\/strong><\/p>\n<p>O per\u00edmetro do trap\u00e9zio \u00e9 $P=16+5+10+5=36\\,cm$.<\/p>\n<p>Como o trap\u00e9zio \u00e9 is\u00f3sceles, ent\u00e3o os tri\u00e2ngulos [ADE] e [BCF] s\u00e3o geometricamente iguais.<br \/>\nLogo, \\[\\overline{AE}=\\overline{BF}=\\frac{\\overline{AB}-\\overline{CD}}{2}=\\frac{16-10}{2}=3\\,cm\\]<\/p>\n<p>Aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [ADE], temos ${{\\overline{AE}}^{2}}+{{\\overline{ED}}^{2}}={{\\overline{AD}}^{2}}$.<br \/>\nLogo:\u00a0\u00a0\\[\\begin{array}{*{35}{l}}<br \/>\n{{3}^{2}}+{{\\overline{ED}}^{2}}={{5}^{2}} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{ED}}^{2}}={{5}^{2}}-{{3}^{2}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{ED}}^{2}}=25-9\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{ED}}^{2}}=16\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{ED}=4\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto,\u00a0a \u00e1rea do trap\u00e9zio \u00e9 \\[A=\\frac{16+10}{2}\\times 4=52\\,c{{m}^{2}}\\]<br \/>\n\u00ad<\/p>\n<p style=\"text-align: left;\"><strong><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12470\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12470\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8b.png\" data-orig-size=\"735,419\" 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=\"Al\u00ednea b)\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8b.png\" class=\"alignright wp-image-12470 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8b-300x171.png\" alt=\"Al\u00ednea b)\" width=\"300\" height=\"171\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8b-300x171.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8b.png 735w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>b)<\/strong><\/p>\n<p>Como o trap\u00e9zio \u00e9 is\u00f3sceles, ent\u00e3o os tri\u00e2ngulos [ADE] e [BCF] s\u00e3o geometricamente iguais.<br \/>\nLogo, \\[\\overline{AE}=\\overline{BF}=\\frac{\\overline{AB}-\\overline{CD}}{2}=\\frac{40,5-20,5}{2}=10\\,cm\\]<\/p>\n<p>Aplicando o teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [AED], temos ${{\\overline{AE}}^{2}}+{{\\overline{ED}}^{2}}={{\\overline{AD}}^{2}}$.<br \/>\nLogo:\u00a0\\[\\begin{array}{*{35}{l}}<br \/>\n{{10}^{2}}+{{14}^{2}}={{\\overline{AD}}^{2}} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{AD}}^{2}}={{10}^{2}}+{{14}^{2}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{AD}}^{2}}=100+196\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{AD}=\\sqrt{296}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Como $\\overline{AD}=\\sqrt{296}\\simeq 17,2\\,cm$, os valores do per\u00edmetro (aproximado)\u00a0e da \u00e1rea do trap\u00e9zio s\u00e3o:<\/p>\n<p>\\[P=40,5+17,2+20,5+17,2=95,4\\,cm\\]<\/p>\n<p>\\[A=\\frac{40,5+20,5}{2}\\times 14=427\\,c{{m}^{2}}\\]<br \/>\n\u00ad<\/p>\n<p style=\"text-align: left;\"><strong><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8c.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12471\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12471\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8c.png\" data-orig-size=\"836,484\" 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=\"Al\u00ednea c)\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8c.png\" class=\"alignright wp-image-12471 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8c-300x174.png\" alt=\"Al\u00ednea c)\" width=\"300\" height=\"174\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8c-300x174.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag-32-8c.png 836w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>c)<\/strong><\/p>\n<p>Como $\\overline{BE}=\\overline{AB}-\\overline{CD}=42-18=24\\,cm$, aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [BCE], temos: ${{\\overline{CE}}^{2}}+{{\\overline{EB}}^{2}}={{\\overline{BC}}^{2}}$.<br \/>\nLogo: \\[\\begin{array}{*{35}{l}}<br \/>\n{{18}^{2}}+{{24}^{2}}={{\\overline{BC}}^{2}} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{BC}}^{2}}={{18}^{2}}+{{24}^{2}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\overline{BC}}^{2}}=324+576\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{BC}=\\sqrt{900}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\overline{BC}=30\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Os valores do per\u00edmetro\u00a0e da \u00e1rea do trap\u00e9zio s\u00e3o:<\/p>\n<p>\\[P=42+30+18+18=108\\,cm\\]<\/p>\n<p>\\[A=\\frac{42+18}{2}\\times 18=540\\,c{{m}^{2}}\\]<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6126' onClick='GTTabs_show(0,6126)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determina o per\u00edmetro e a \u00e1rea de cada um dos seguintes trap\u00e9zios (as medidas est\u00e3o em cent\u00edmetros): Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20673,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,112],"tags":[424,108,67,118],"series":[],"class_list":["post-6126","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-decomposicao-de-figuras-teorema-de-pitagoras","tag-8-o-ano","tag-area","tag-geometria","tag-teorema-de-pitagoras"],"views":4735,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/8V1Pag032-8_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6126","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=6126"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6126\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20673"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6126"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}