{"id":5110,"date":"2025-06-07T12:48:00","date_gmt":"2025-06-07T16:48:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5110"},"modified":"2025-08-18T12:49:48","modified_gmt":"2025-08-18T16:49:48","slug":"geometry-challenge-2025-06-07","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5110","title":{"rendered":"Geometry Challenge &#8211; 2025\/06\/07"},"content":{"rendered":"\n<p>Two opposite edges of a unit square are folded along one of the diagonal to form a parallelogram. Then, those two opposite edges on the parallelogram are folded together to form a trapezoid. Find the area of the trapezoid.<a href=\"javascript:toggle_visibility('sol_geo_chall_2025_06_07')\">\ud83d\udd11<\/a><\/p>\n<div id=\"sol_geo_chall_2025_06_07\" style=\"display:none\">\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/08\/Screenshot-2025-08-18-at-12.12.11\u202fPM.png\" alt=\"\" class=\"wp-image-5112\" width=\"300\" height=\"300\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/08\/Screenshot-2025-08-18-at-12.12.11\u202fPM.png 704w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/08\/Screenshot-2025-08-18-at-12.12.11\u202fPM-300x300.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2025\/08\/Screenshot-2025-08-18-at-12.12.11\u202fPM-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure>\n\n\n\n<p><strong>Solution<\/strong>: Edge $AD$ and $CB$ of unit square $ABCD$ are folded along diagonal $AC$, forming parallelogram $AECF$. Paralleogram $AECF$ is folded along $GH$, forming trapezoid $AIJB$. <\/p>\n\n\n\n<p>Let $x=BE$, Because $B&#8217;E=BE$, and $\\triangle{AB&#8217;E}$ is an isosceles right triangle, and $AE=AB-BE=1-x$, by Pythagorean Theorem, we have $$x^2+x^2=(1-x)^2$$<\/p>\n\n\n\n<p>Solving the above equation and ignoring the invalid $x$ value, we have $x=\\sqrt{2}-1$. Therefore <\/p>\n\n\n\n<p>$$Area(\\triangle{EBC})=\\dfrac{1}{2}\\cdot BE\\cdot CB=\\dfrac{1}{2}\\cdot (\\sqrt{2}-1)\\cdot 1=\\dfrac{\\sqrt{2}-1}{2}$$<\/p>\n\n\n\n<p>$$Area(\\triangle{BHJ})=\\dfrac{1}{4}Area(\\triangle{EBC})=\\dfrac{1}{4}\\cdot\\dfrac{\\sqrt{2}-1}{2}=\\dfrac{\\sqrt{2}-1}{8}$$<\/p>\n\n\n\n<p>$$Area(AIJB)=Area(AGHB)-Area(\\triangle{AGI})-Area(\\triangle{BHJ})$$<\/p>\n\n\n\n<p>$$=\\dfrac{1}{2}-2\\cdot Area(\\triangle{BHJ})=\\dfrac{1}{2}-2\\cdot\\dfrac{\\sqrt{2}-1}{8}=\\boxed{\\dfrac{3-\\sqrt{2}}{4}}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Two opposite edges of a unit square are folded along one of the diagonal to form a parallelogram. Then, those two opposite edges on the parallelogram are folded together to form a trapezoid. Find the area of the trapezoid.\ud83d\udd11 Solution: &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5110\">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":[9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5110"}],"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=5110"}],"version-history":[{"count":14,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5110\/revisions"}],"predecessor-version":[{"id":5127,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5110\/revisions\/5127"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5110"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5110"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}