{"id":141,"date":"2020-01-26T15:30:05","date_gmt":"2020-01-26T15:30:05","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=141"},"modified":"2020-10-12T15:34:22","modified_gmt":"2020-10-12T15:34:22","slug":"mathcounts-exercises-16","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=141","title":{"rendered":"MATHCOUNTS Exercises &#8211; 16"},"content":{"rendered":"\n<p style=\"text-align: left;\">A semicircle and a circle are placed inside a square with sides of length 4 cm as shown. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a side of the square. In centimeters, what is the radius of the circle?<a href=\"wp-content\/uploads\/2020\/10\/2012-state-target-6.png\"><img width=\"200\" src=\"wp-content\/uploads\/2020\/10\/2012-state-target-6.png\" alt=\"\" class=\"aligncenter wp-image-143\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/2012-state-target-6.png 448w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/2012-state-target-6-300x257.png 300w\" sizes=\"(max-width: 448px) 100vw, 448px\" \/><\/a><em>Source: MATHCOUNTS 2012 State Target Round<\/em>. <a onclick=\"toggle_visibility('mathcounts-exercises-16-solution');\">Click here for the solution.<\/a><\/p>\n<div id=\"mathcounts-exercises-16-solution\" style=\"display:none;\">\n<p><strong>Solution<\/strong> First, connect the center of the circle with the center of the semicircle. Then drop a perpendicular from the center of the circle to side AB, as shown.<\/p>\n<a href=\"wp-content\/uploads\/2020\/10\/2012-state-target-6-B.jpg\"><img width=\"224\" src=\"wp-content\/uploads\/2020\/10\/2012-state-target-6-B.jpg\" alt=\"\" class=\"aligncenter wp-image-142\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/2012-state-target-6-B.jpg 415w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/2012-state-target-6-B-296x300.jpg 296w\" sizes=\"(max-width: 415px) 100vw, 415px\" \/><\/a>\n<p>Let \\(r\\) be the radius of the circle. So, segment \\(IJ=2+r\\), since the side of square \\(ABCD\\) is 4. Also, segment \\(IK=4-r\\) and \\(KJ=2-r\\). Since \\(IK \\perp KJ\\), we can set up a Pythagorean:<br \/>\n$$(4-r)^2+(2-r)^2=(2+r)^2$$<br \/>\nThis simplifies to $$16-8r+r^2+4-4r+r^2=4+4r+r^2$$ Simplifying further, we get a quadratic equation:<br \/>\n$$r^2-16r+16=0$$<br \/>\nApplying the quadratic formula, we get \\(r=8-4\\sqrt{3}\\).<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A semicircle and a circle are placed inside a square with sides of length 4 cm as shown. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=141\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[12,9,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/141"}],"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=141"}],"version-history":[{"count":4,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/141\/revisions"}],"predecessor-version":[{"id":147,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/141\/revisions\/147"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}