{"id":2237,"date":"2021-10-16T20:19:16","date_gmt":"2021-10-16T20:19:16","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2237"},"modified":"2024-10-25T12:09:50","modified_gmt":"2024-10-25T16:09:50","slug":"geometry-challenge-8-%e2%ad%90","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2237","title":{"rendered":"Geometry Challenge &#8211; 8 \u2b50"},"content":{"rendered":"\n<p>In unit square $ABCD$, point $E$ and $F$ are located on edge $CD$, with $E$ closer to $D$ and $F$ closer to $C$. Line $BE$ and $AF$ intersect at $G$, forming two triangles: $\\triangle{ABG}$ and $\\triangle{EFG}$. Find the minimum value of the total area formed by these two triangles. Click here for the <a href=\"javascript:toggle_visibility('sol-geo-chall-8');\">solution<\/a>.<\/p>\n\n\n\n<div id=\"sol-geo-chall-8\" style=\"display:none\">\n\n\n\n<p>Solution: Draw line $PQ$ passing thru $G$ and parallel to $AD$, intersecting $AB$ and $CD$ at $P$ and $Q$ respectively. Since $ABCD$ is a unit square, the total area of $\\triangle{ABG}$ and $\\triangle{EFG}$ is $$S=\\dfrac{1}{2}(AB\\cdot GP+CD\\cdot DE)=\\dfrac{1}{2}(GP+CD\\cdot(PQ-GP))$$ $$=\\dfrac{1}{2}(GP+CD\\cdot(1-GP))$$<\/p>\n\n\n\n<p>Because $\\triangle{ABG}\\sim\\triangle{EFG}$, we have $$\\dfrac{AB}{GP}=\\dfrac{CD}{GQ}$$ Therefore, $$CD=\\dfrac{AB\\cdot GQ}{CP}=\\dfrac{GQ}{GP}=\\dfrac{1-GP}{GP}$$<\/p>\n\n\n\n<p>We have $$S=\\dfrac{1}{2}(GP+\\dfrac{(1-GP)^2}{GP})=GP+\\dfrac{1}{2\\cdot GP}-1$$ $$\\ge 2\\cdot\\sqrt{GP\\cdot \\dfrac{1}{2\\cdot GP}}-1=\\sqrt{2}-1$$<\/p>\n\n\n\n<p>Therefore the minimum value of $S$ is $\\boxed{\\sqrt{2}-1}$, when $GP=\\dfrac{1}{2\\cdot GP}$, i.e. $GP=\\dfrac{\\sqrt{2}}{2}$, and $CD=\\sqrt{2}-1$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>In unit square $ABCD$, point $E$ and $F$ are located on edge $CD$, with $E$ closer to $D$ and $F$ closer to $C$. Line $BE$ and $AF$ intersect at $G$, forming two triangles: $\\triangle{ABG}$ and $\\triangle{EFG}$. Find the minimum value &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2237\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,14,9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2237"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2237"}],"version-history":[{"count":19,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2237\/revisions"}],"predecessor-version":[{"id":2262,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2237\/revisions\/2262"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}