{"id":2773,"date":"2025-02-10T12:35:02","date_gmt":"2025-02-10T10:35:02","guid":{"rendered":"https:\/\/matematicastro.es\/?p=2773"},"modified":"2025-02-10T15:01:51","modified_gmt":"2025-02-10T13:01:51","slug":"un-problema-de-trigonometria","status":"publish","type":"post","link":"https:\/\/matematicastro.es\/?p=2773","title":{"rendered":"Un problema de trigonometr\u00eda"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><\/p>\n<\/blockquote>\n\n\n\n<p class=\"has-text-align-right has-text-color has-link-color wp-elements-24b49903180e1701448cc3970b74b750\" style=\"color:#5e1e1e;font-size:16px\"><em>Son capaces, porque creen que son capaces<\/em><\/p>\n\n\n\n<p class=\"has-text-align-right has-text-color has-link-color wp-elements-6591b7343628c06886e97d9d1d3523f2\" style=\"color:#5e1e1e;font-size:16px\"><em><a href=\"https:\/\/es.wikipedia.org\/wiki\/Virgilio\" data-type=\"link\" data-id=\"https:\/\/es.wikipedia.org\/wiki\/Virgilio\" target=\"_blank\" rel=\"noreferrer noopener\">Virgilio<\/a><\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-alpha-channel-opacity has-background is-style-default\" style=\"background-color:#5e1e1e;color:#5e1e1e\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Enunciado<\/h4>\n\n\n\n<p>En un rect\u00e1ngulo $ABCD$ de lados 8 cm y 12 cm, se traza desde $B$ una perpendicular a la diagonal $AC$, y desde $D$, otra perpendicular a la misma diagonal. Sean $M$ y $N$ los puntos donde estas perpendiculares cortan a la diagonal. Halla la longitud del segmento $MN$ (ver figura).<\/p>\n\n\n<div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"607\" height=\"292\" src=\"https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/rectangulo.jpg\" alt=\"\" class=\"wp-image-2774\" style=\"object-fit:cover;width:439px;height:auto\" srcset=\"https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/rectangulo.jpg 607w, https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/rectangulo-300x144.jpg 300w\" sizes=\"auto, (max-width: 607px) 100vw, 607px\" \/><\/figure>\n<\/div>\n\n\n<h4 class=\"wp-block-heading\">Soluci\u00f3n<\/h4>\n\n\n\n<p>La idea es hallar la diagonal $AC$ y luego hallar $AN=MC$. De este modo tendremos que $MN=AC-2AN$.<\/p>\n\n\n\n<p>En el tri\u00e1ngulo $ABC$, que es rect\u00e1ngulo en $B$, por el teorema de Pit\u00e1goras, se tiene (ver figura):<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"626\" height=\"287\" src=\"https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/triangulo1.jpg\" alt=\"\" class=\"wp-image-2775\" style=\"width:373px;height:auto\" srcset=\"https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/triangulo1.jpg 626w, https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/triangulo1-300x138.jpg 300w\" sizes=\"auto, (max-width: 626px) 100vw, 626px\" \/><\/figure>\n<\/div>\n\n\n<p>$$AC^2=12^2+8^2\\Rightarrow AC^2=208\\Rightarrow AC\\cong14,42\\,\\text{cm}$$<\/p>\n\n\n\n<p>En el mismo tri\u00e1ngulo anterior tambi\u00e9n podemos hallar el \u00e1ngulo $\\alpha$, haciendo uso de la tangente:<\/p>\n\n\n\n<p>$$\\text{tg}\\,\\alpha=\\frac{8}{12}\\Rightarrow\\alpha=\\text{arctg}\\frac{8}{12}\\Rightarrow\\alpha=33,69\u00ba$$ <\/p>\n\n\n\n<p>Para hallar $AN$ vamos a usar el tri\u00e1ngulo $AND$, que es rect\u00e1ngulo en $N$ (ver figura siguiente).<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"291\" src=\"https:\/\/matematicastro.es\/wp-content\/uploads\/2025\/02\/triangulo2.jpg\" alt=\"\" class=\"wp-image-2780\" style=\"width:161px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>En \u00e9l, est\u00e1 claro que $\\beta=90\u00ba-\\alpha\\Rightarrow\\beta=90\u00ba-33,69\u00ba\\Rightarrow\\beta=56,31\u00ba$. Conv\u00e9ncete observando el rect\u00e1ngulo inicial y los dos tri\u00e1ngulos anteriores.<\/p>\n\n\n\n<p>Entonces:<\/p>\n\n\n\n<p>$$\\cos \\beta = \\frac{{AN}}{8} \\Rightarrow AN = 8\\cos 56,31 \\Rightarrow AN = 4,44\\,\\text{cm}$$<\/p>\n\n\n\n<p>Finalmente:<\/p>\n\n\n\n<p>$$MN = AC &#8211; 2AN = 14,42 &#8211; 2 \\cdot 4,44 \\Rightarrow MN = 5,54\\,\\text{cm}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Son capaces, porque creen que son capaces Virgilio Enunciado En un rect\u00e1ngulo $ABCD$ de lados 8 cm y 12 cm, se traza desde $B$ una perpendicular a la diagonal $AC$, y desde $D$, otra perpendicular a la misma diagonal. Sean $M$ y $N$ los puntos donde estas perpendiculares cortan a la diagonal. Halla la longitud [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2774,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-2773","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","6":"hentry","7":"category-uncategorized","9":"post-with-thumbnail","10":"post-with-thumbnail-large"},"_links":{"self":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/2773","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2773"}],"version-history":[{"count":23,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/2773\/revisions"}],"predecessor-version":[{"id":2851,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/2773\/revisions\/2851"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/media\/2774"}],"wp:attachment":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2773"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2773"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}