{"id":183,"date":"2019-12-21T20:57:17","date_gmt":"2019-12-21T20:57:17","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=183"},"modified":"2020-10-16T13:29:41","modified_gmt":"2020-10-16T13:29:41","slug":"mathcounts-exercises-1","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=183","title":{"rendered":"MATHCOUNTS Exercises &#8211; 1"},"content":{"rendered":"\n<p><strong>MATHCOUNTS 2016-2017 \u2013 231<\/strong> Regular nonagon ABCDEFGHI is inscribed in a circle, as shown. What is $m\\angle{AHC}$? <a href=\"javascript:toggle_visibility('sol-mathcounts-exercises-1');toggle_visibility('img-mathcounts-exercises-1');\">Click here to show the solutions.<\/a><\/p>\n\n\n\n<div id=\"img-mathcounts-exercises-1\" style=\"display:block\">\n<img width=\"200\" src=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-PM.png\" alt=\"\" class=\"wp-image-184\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-PM.png 576w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-PM-300x300.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-PM-150x150.png 150w\" sizes=\"(max-width: 576px) 100vw, 576px\" \/>\n<\/div>\n\n\n\n<div id=\"sol-mathcounts-exercises-1\" style=\"display:none\">\n\n\n\n<img width=\"200\" src=\"wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-1PM-1.png\" alt=\"\" class=\"wp-image-191\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-1PM-1.png 576w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-1PM-1-300x300.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2020-10-12-at-4.25.31-1PM-1-150x150.png 150w\" sizes=\"(max-width: 576px) 100vw, 576px\" \/>\n<p><strong>Solution 1<\/strong> Draw line $\\overline{AC}$. \n$$\n\\begin{align}\n&amp;\\because\\ ABCDEFGHI\\ \\text{is a regular nonagon} \\\\\n&amp;\\therefore \\triangle{ABC}, \\triangle{AIH}, \\triangle{ACH} \\text{are isosceles triangle} \\\\ \n&amp;\\therefore m\\angle{HIA}=m\\angle{IAB}=m\\angle{ABC}=\\dfrac{(9-2)\\cdot 180^\\circ}{9}=140^\\circ\\\\\n&amp;\\because \\triangle{HIA} \\cong \\triangle{ABC} \\\\\n&amp;\\therefore m\\angle{IAH}=m\\angle{BAC}=\\dfrac{180^\\circ-m\\angle{ABC}}{2}=\\dfrac{180^\\circ-140^\\circ}{2}=20^\\circ \\\\\n&amp;\\therefore m\\angle{CAH}=m\\angle{IAB}-m\\angle{IAH}-m\\angle{BAC}=140^\\circ-20^\\circ-20^\\circ=100^\\circ \\\\\n&amp;\\therefore m\\angle{AHC}=\\dfrac{180^\\circ-m\\angle{CAH}}{2}=\\dfrac{180^\\circ-100^\\circ}{2}=40^\\circ \\\\\n\\end{align}\n$$\n<\/p>\n<p><strong>Solution 2<\/strong>\n$$\n\\begin{align}\n&amp;\\because\\ ABCDEFGHI\\ \\text{is a regular nonagon} \\\\\n&amp;\\therefore m\\overparen{AB}=m\\overparen{BC}=\\dfrac{180^\\circ}{9}=20^\\circ \\\\\n&amp;\\therefore m\\triangle{AMC}=m\\overparen{AB}=m\\overparen{BC}=20^\\circ+20^\\circ=40^\\circ \\\\\n\\end{align} \\\\\n$$\n<\/p>\n\n\n\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>MATHCOUNTS 2016-2017 \u2013 231 Regular nonagon ABCDEFGHI is inscribed in a circle, as shown. What is $m\\angle{AHC}$? Click here to show the solutions. Solution 1 Draw line $\\overline{AC}$. $$ \\begin{align} &amp;\\because\\ ABCDEFGHI\\ \\text{is a regular nonagon} \\\\ &amp;\\therefore \\triangle{ABC}, \\triangle{AIH}, &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=183\">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\/183"}],"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=183"}],"version-history":[{"count":23,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/183\/revisions"}],"predecessor-version":[{"id":477,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/183\/revisions\/477"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=183"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=183"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}