{"id":3525,"date":"2023-11-30T19:52:16","date_gmt":"2023-11-30T23:52:16","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=3525"},"modified":"2024-10-25T09:57:50","modified_gmt":"2024-10-25T13:57:50","slug":"geometry-challenge-14","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=3525","title":{"rendered":"Geometry Challenge &#8211; 14"},"content":{"rendered":"\n<p>In the figure as shown, $ABCD$ is a unit square, $E$ is on $BC$, $F$ is on $CD$, $\\angle{EAF}=45^\\circ$, $\\angle{AEB}=70^\\circ$. <\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized alignright\">\n<div id=\"geo-chall-14\" name=\"geo-chall-14\" style=\"display:block\">\n<svg width=\"180\" height=\"140\">\n<g transform=\"translate(40,0)\">\n<polygon points=\"20,20 20,120 120,120 120,20\" fill=\"none\" stroke=\"black\"\/>\n<line x1=\"20\" y1=\"20\" x2=\"66\" y2=\"120\" stroke=\"black\"\/>\n<line x1=\"20\" y1=\"20\" x2=\"120\" y2=\"56\" stroke=\"black\"\/>\n<line x1=\"120\" y1=\"56\" x2=\"66\" y2=\"120\" stroke=\"black\"\/>\n\n<text x=\"20\" y=\"17\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">A<\/text>\n<text x=\"120\" y=\"17\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">B<\/text>\n<text x=\"120\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">C<\/text>\n<text x=\"20\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">D<\/text>\n\n<text x=\"128\" y=\"60\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">E<\/text>\n<text x=\"66\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">F<\/text>\n\n<path d=\"M 120 45 A 10 10 0 0 0 110 53\" stroke=\"black\" fill=\"none\"\/>\n<text x=\"108\" y=\"40\" text-anchor=\"middle\" font-size=\"0.75em\" stroke=\"black\">70\u00b0<\/text>\n<path d=\"M 31 24 A 10 10 0 0 1 24.5 31\" stroke=\"black\" fill=\"none\"\/>\n<text x=\"40\" y=\"40\" text-anchor=\"middle\" font-size=\"0.75em\" stroke=\"black\">45\u00b0<\/text>\n<path d=\"M 66 120 L 61.75 111 A 10 10 0 0 1 72.5 112.5 L 66 120\" stroke=\"black\" fill=\"grey\"\/>\n<text x=\"68\" y=\"107\" text-anchor=\"middle\" font-size=\"0.75em\" stroke=\"black\">?<\/text>\n<\/svg>\n<\/g>\n<\/div>\n<div id=\"geo-chall-14-sol\" name=\"geo-chall-14-sol\" style=\"display:none\">\n<figure class=\"wp-block-image size-large is-resized alignright\">\n<svg width=\"180\" height=\"140\">\n<g transform=\"translate(40,0)\">\n<polygon points=\"20,20 20,120 120,120 120,20\" fill=\"none\" stroke=\"black\"\/>\n<line x1=\"20\" y1=\"20\" x2=\"66\" y2=\"120\" stroke=\"black\"\/>\n<line x1=\"20\" y1=\"20\" x2=\"120\" y2=\"56\" stroke=\"black\"\/>\n<line x1=\"120\" y1=\"56\" x2=\"66\" y2=\"120\" stroke=\"black\"\/>\n<line x1=\"20\" y1=\"120\" x2=\"-16\" y2=\"120\" stroke=\"red\"\/>\n<line x1=\"20\" y1=\"20\" x2=\"-16\" y2=\"120\" stroke=\"red\"\/>\n\n<text x=\"20\" y=\"17\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">A<\/text>\n<text x=\"120\" y=\"17\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">B<\/text>\n<text x=\"120\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">C<\/text>\n<text x=\"20\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">D<\/text>\n\n<text x=\"128\" y=\"60\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">E<\/text>\n<text x=\"66\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">F<\/text>\n<text x=\"-16\" y=\"133\" text-anchor=\"middle\" font-size=\"smaller\" stroke=\"black\">G<\/text>\n\n<path d=\"M 120 45 A 10 10 0 0 0 110 53\" stroke=\"black\" fill=\"none\"\/>\n<text x=\"108\" y=\"40\" text-anchor=\"middle\" font-size=\"0.75em\" stroke=\"black\">70\u00b0<\/text>\n<path d=\"M 31 24 A 10 10 0 0 1 24.5 31\" stroke=\"black\" fill=\"none\"\/>\n<text x=\"40\" y=\"40\" text-anchor=\"middle\" font-size=\"0.75em\" stroke=\"black\">45\u00b0<\/text>\n<path d=\"M 66 120 L 61.75 111 A 10 10 0 0 1 72.5 112.5 L 66 120\" stroke=\"black\" fill=\"grey\"\/>\n<text x=\"68\" y=\"107\" text-anchor=\"middle\" font-size=\"0.75em\" stroke=\"black\">?<\/text>\n<\/svg>\n<\/g>\n<\/div>\n<\/figure>\n\n\n\n<p>(1) Find the value of $\\angle{AFE}.$<\/p>\n\n\n\n<p>(2) Find the perimeter of $\\triangle{CEF}$.<\/p>\n\n\n\n<p>Click <a href=\"javascript:toggle_visibility('geo-chall-14');toggle_visibility('geo-chall-14-sol');toggle_visibility('geo-chall-14-sol1');\">here<\/a> for the solution.<\/p>\n<div id=\"geo-chall-14-sol1\" name=\"geo-chall-14-sol1\" style=\"display:none\">\n\n\n\n<p><strong>Solution:<\/strong> Rotating $\\triangle{ABE}$ $90^\\circ$ degree clockwise around $A$, we have $\\triangle{AFE}\\cong\\triangle{AEG}$. $\\angle{AFE}=65^\\circ$ and the perimeter of $\\triangle{CEF}$ is $2$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>In the figure as shown, $ABCD$ is a unit square, $E$ is on $BC$, $F$ is on $CD$, $\\angle{EAF}=45^\\circ$, $\\angle{AEB}=70^\\circ$. A B C D E F 70\u00b0 45\u00b0 ? A B C D E F G 70\u00b0 45\u00b0 ? 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