{"id":5416,"date":"2010-11-14T01:44:44","date_gmt":"2010-11-14T01:44:44","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5416"},"modified":"2022-01-12T01:59:21","modified_gmt":"2022-01-12T01:59:21","slug":"rascunho-27","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5416","title":{"rendered":"Averigue se s\u00e3o ou n\u00e3o perpendiculares as retas $r$ e $s$"},"content":{"rendered":"<p><ul id='GTTabs_ul_5416' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5416' class='GTTabs_curr'><a  id=\"5416_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5416' ><a  id=\"5416_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_5416'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Averigue se s\u00e3o ou n\u00e3o perpendiculares as retas r e s de equa\u00e7\u00f5es:<\/p>\n<ol>\n<li>r: $y=2x-3$ e s: $y=-x+\\frac{1}{2}$;<\/li>\n<li>r: $x=3$ e s: $y=4$;<\/li>\n<li>r: $2x+3y-1=0$ e s: $3x-2y+7=0$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5416' onClick='GTTabs_show(1,5416)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5416'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>O declive da reta r \u00e9 ${{m}_{r}}=2$ e o declive da reta s \u00e9 ${{m}_{s}}=-1$.<br \/>\nLogo, as retas n\u00e3o s\u00e3o perpendiculares, pois o declive de uma n\u00e3o \u00e9 sim\u00e9trico do inverso do da outra: $2\\ne -\\frac{1}{-1}$.<br \/>\n\u00ad<\/li>\n<li>A reta r \u00e9 paralela ao eixo Oy e a reta s \u00e9 paralela ao eixo Ox.<br \/>\nLogo as retas r e s s\u00e3o perpendiculares.<br \/>\n\u00ad<\/li>\n<li>Como \\(2x + 3y &#8211; 1 = 0 \\Leftrightarrow y = &#8211; \\frac{2}{3}x + \\frac{1}{3}\\), ent\u00e3o ${{m}_{r}}=-\\frac{2}{3}$;<br \/>\n$3x-2y+7=0\\Leftrightarrow y=\\frac{3}{2}x+\\frac{7}{2}$, logo ${{m}_{s}}=\\frac{3}{2}$.<br \/>\nPortanto, as retas r e s s\u00e3o perpendiculares, pois o declive de uma delas \u00e9 sim\u00e9trico do inverso do declive da outra: $-\\frac{2}{3}=-\\frac{1}{\\frac{3}{2}}$.<br \/>\n\u00ad<\/li>\n<\/ol>\n<blockquote>\n<p><strong>Uma reta r de equa\u00e7\u00e3o $y=mx+b$ admite como vetor diretor o vetor $\\vec{r}(1,m)$ e como vetor normal $\\vec{n}(-m,1)$ .<\/strong><\/p>\n<p><strong>Se $m\\ne 0$, ent\u00e3o $-\\frac{1}{m}$ \u00e9 o declive das retas perpendiculares a r.<\/strong><\/p>\n<p><strong>Isto \u00e9, retas perpendiculares n\u00e3o paralelas aos eixos coordenados possuem declives tais que: o declive de uma das retas \u00e9 sim\u00e9trico do inverso do declive da outra reta, ou seja, $m&#8217;=-\\frac{1}{m}$.<\/strong><\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5416' onClick='GTTabs_show(0,5416)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Averigue se s\u00e3o ou n\u00e3o perpendiculares as retas r e s de equa\u00e7\u00f5es: r: $y=2x-3$ e s: $y=-x+\\frac{1}{2}$; r: $x=3$ e s: $y=4$; r: $2x+3y-1=0$ e s: $3x-2y+7=0$. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19189,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5416","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","tag-vectores"],"views":2785,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat75.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5416","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=5416"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5416\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5416"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5416"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5416"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5416"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}