{"id":2555,"date":"2022-02-21T02:01:00","date_gmt":"2022-02-21T06:01:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2555"},"modified":"2024-10-25T11:15:56","modified_gmt":"2024-10-25T15:15:56","slug":"geometry-challenge-9","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2555","title":{"rendered":"Geometry Challenge \u2013 9"},"content":{"rendered":"\n<p>In the diagram below, two tangent semi-circles centered at A and B are inscribed in a large semi-circle center at O, and circle C is tangent to semi-circle A, B and O. Point A, O, and B are on the same line. Given the radii of semi-circle A and B as 3 and 2 respectively, the radius of circle C can be expressed as $\\dfrac{m}{n}$, where $m$ and $n$ are co-prime. Find the value of $m+n$.<\/p>\n\n\n\n<center>\n<svg width=\"230\" height=\"140\">\n<g transform=\"translate(20,0)\">\n<path d=\"M 0,120 A 100,100 1,0 1 200,120 L 0,120\" fill=\"none\" stroke=\"black\"><\/path>\n<path d=\"M 0,120 A 60,60 1,0 1 120,120 A 40,40 1,0 1 200,120 L 0,120\" fill=\"white\" stroke=\"black\"><\/path>\n<circle cx=\"126\" cy=\"57\" stroke=\"black\" fill=\"none\" r=\"31.5\"><\/circle>\n<circle cx=\"126\" cy=\"57\" stroke=\"black\" fill=\"black\" r=\"1\"><\/circle>\n<circle cx=\"60\" cy=\"120\" stroke=\"black\" fill=\"black\" r=\"1\"><\/circle>\n<circle cx=\"160\" cy=\"120\" stroke=\"black\" fill=\"black\" r=\"1\"><\/circle>\n<circle cx=\"100\" cy=\"120\" stroke=\"black\" fill=\"black\" r=\"1\"><\/circle>\n\n<text x=\"100\" y=\"135\" text-anchor=\"middle\">O<\/text>\n<text x=\"126\" y=\"48\" text-anchor=\"middle\">C<\/text>\n<text x=\"60\" y=\"135\" text-anchor=\"middle\">A<\/text>\n<text x=\"160\" y=\"135\" text-anchor=\"middle\">B<\/text>\n<\/g>\n<\/svg>\n<\/center>\n\n\n\n<p><em>Hints: Solution-1: Cosine law; Solution-2: Heron Formula; Solution-3: Descartes&#8217; Theorem.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the diagram below, two tangent semi-circles centered at A and B are inscribed in a large semi-circle center at O, and circle C is tangent to semi-circle A, B and O. Point A, O, and B are on the &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2555\">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":[13,9,15],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2555"}],"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=2555"}],"version-history":[{"count":4,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2555\/revisions"}],"predecessor-version":[{"id":2559,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2555\/revisions\/2559"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2555"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2555"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2555"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}