{"id":4867,"date":"2025-02-09T13:37:00","date_gmt":"2025-02-09T17:37:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4867"},"modified":"2025-03-08T13:10:34","modified_gmt":"2025-03-08T17:10:34","slug":"geometry-challenge-18","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4867","title":{"rendered":"Geometry Challenge – 18"},"content":{"rendered":"\n

Two squares $ABCD$ and $DEFG$ are inscribed inside a unit semi-circle, as shown in the following diagram, with $CD$ and $DE$ on the same line, $A$, $D$, $G$ on the diameter of the semi-circle, and $B$ and $F$ on the semi-circle. (1) Find the sum of areas of two squares. (2) Find the smallest area ratio between the two squares. Click here<\/a> for the solution.<\/p>\n\n\n\n

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(1) <\/strong>Without loss of generality, assume $AB\\le DE$. Select $O$ between $D$ and $E$ so that $OD=CE$. And draw line $DB$, $FF$, $BF$, $OB$, and $OF$, as shown below:<\/p>\n\n\n\n

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Because $OG=DG-OD=DE-CE=CD=AB$, $AO=AD+OD=DC+CE=DE=GF$, and $\\angle{BAD}=\\angle{FGD}=90^\\circ$, $\\triangle{BAO}\\cong\\triangle{OGF}$. Therefore, $OB=OF$.<\/p>\n\n\n\n

Additionally, because $\\angle{BOF}=180^\\circ-\\angle{AOB}-\\angle{FOG}=180^\\circ-(\\angle{AOB}+\\angle{ABO})=180^\\circ-90^\\circ=90^\\circ$. Therefore, $\\triangle{BOF}$ is an isosceles right triangle. $O$ is the center of the semi-triangle. And $OB=OF=1$, $BF=\\sqrt{2}$.<\/p>\n\n\n\n

Becaue $\\angle{ADB}=\\angle{GDF}=45^\\circ$, we have $\\angle{BDF}=180^\\circ -(\\angle{ADB}+\\angle{GDF})=180^\\circ-90^\\circ=90^\\circ$.<\/p>\n\n\n\n

Therefore, $\\triangle{BDF}$ is right. We have $BD^2+DF^2=BF^2=2$.<\/p>\n\n\n\n

So the sum of the areas of two squares is: $$AB^2+DE^2=\\dfrac{1}{2}BD^2+\\dfrac{1}{2}DF^2=\\dfrac{1}{2}(BD^2+DF^2)=\\dfrac{1}{2}\\cdot 2=\\boxed{1}$$<\/p>\n\n\n\n

(2)<\/strong> Without loss of generality, assume $AB\\le DE$. In order to maximize the difference between the area of $DEFG$ and $ABCD$, $EF$ must be as long as possible, which puts both $E$ and $F$ on the semi-circle, as shown below:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore, $OD=OG=\\dfrac{1}{2}EF$, as $EF$ is a chord of the semi-circle. We have $DE^2+OD^2=EO^2$, i.e. $EF^2+\\dfrac{1}{4}EF^2=1$. Therefore, $EF^2=\\dfrac{4}{5}$. <\/p>\n\n\n\n

Hence, the area of $DEFG$ is $\\dfrac{4}{5}$, and that of $ABCD$ is $1-\\dfrac{4}{5}=\\dfrac{1}{5}$. This results in the smallest area ratio between the two squares as $\\boxed{\\dfrac{1}{4}}$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Two squares $ABCD$ and $DEFG$ are inscribed inside a unit semi-circle, as shown in the following diagram, with $CD$ and $DE$ on the same line, $A$, $D$, $G$ on the diameter of the semi-circle, and $B$ and $F$ on the … Continue reading →<\/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\/4867"}],"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=4867"}],"version-history":[{"count":14,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4867\/revisions"}],"predecessor-version":[{"id":4887,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4867\/revisions\/4887"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4867"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}