{"id":170,"date":"2019-12-22T19:39:13","date_gmt":"2019-12-22T19:39:13","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=170"},"modified":"2020-10-16T13:40:41","modified_gmt":"2020-10-16T13:40:41","slug":"mathcounts-exercises-2","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=170","title":{"rendered":"MATHCOUNTS Exercises &#8211; 2"},"content":{"rendered":"\n<p><strong>MASSCOUNTS 2016-2017 &#8211; 232<\/strong>\nIn Circle A, shown here, $\\overleftrightarrow{BD}$ is tangent to the circle at B, and major $\\overparen{BC\\ }$ has measure $230^\\circ$. What is $m\\angle{CBD}$?\n<a onclick=\"toggle_visibility('solution-masscounts-232');\">Click here for the solution.<\/a><\/p>\n<a href=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-22-at-3.52.18-PM.png\"><img loading=\"lazy\" src=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-22-at-3.52.18-PM.png\" alt=\"\" title=\"Screen Shot 2019-12-22 at 3.52.18 PM\" width=\"260\" height=\"197\" class=\"aligncenter size-full wp-image-171\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-22-at-3.52.18-PM.png 520w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-22-at-3.52.18-PM-300x227.png 300w\" sizes=\"(max-width: 260px) 100vw, 260px\" \/><\/a>\n\n<div id=\"solution-masscounts-232\" name=\"solution-masscounts-232\" style=\"display:none\">\n<p><strong>Solution<\/strong> Draw lines $\\overline{AB}$ and $\\overline{AC}$\n$$\\begin{align}\n&#038;\\because major\\ \\overparen{BC\\ }=230^\\circ \\\\\n&#038;\\therefore minor\\ \\overparen{BC\\ }=360^\\circ-major\\ \\overparen{BC\\ }=360^\\circ-230^\\circ=130^\\circ \\\\\n&#038;\\therefore m\\angle{BAC}=minor\\ \\overparen{BC\\ }=130^\\circ \\\\\n&#038;\\because\\ \\overline{AB} = \\overline{AC} \\\\\n&#038;\\therefore \\triangle{ABC}\\ is\\ isosceles \\\\\n&#038;\\therefore m\\angle{ABC}=m\\angle{ACB}=\\dfrac{180^\\circ-m\\angle{BAC}}{2}=\\dfrac{180^\\circ-130^\\circ}{2}=25^\\circ \\\\\n&#038;\\because\\ \\overleftrightarrow{BD}\\ is\\ tangent\\ to\\ circle\\ A\\ at\\ B \\\\\n&#038;\\therefore\\ \\overline{AB} \\perp \\overleftrightarrow{BD} \\\\\n&#038;\\therefore m\\angle{ABD}=90^\\circ \\\\\n&#038;\\therefore m\\angle{ABC}+m\\angle{CBD}=m\\angle{ABD}=90^\\circ \\\\\n&#038;\\therefore m\\angle{CBD}=90^\\circ-m\\angle{ABC}=90^\\circ-25^\\circ=65^\\circ \\blacksquare \\\\\n\\end{align}\n$$\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>MASSCOUNTS 2016-2017 &#8211; 232 In Circle A, shown here, $\\overleftrightarrow{BD}$ is tangent to the circle at B, and major $\\overparen{BC\\ }$ has measure $230^\\circ$. What is $m\\angle{CBD}$? Click here for the solution. Solution Draw lines $\\overline{AB}$ and $\\overline{AC}$ $$\\begin{align} &#038;\\because &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=170\">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":[9,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/170"}],"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=170"}],"version-history":[{"count":9,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/170\/revisions"}],"predecessor-version":[{"id":481,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/170\/revisions\/481"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=170"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=170"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}