{"id":164,"date":"2019-12-23T18:49:02","date_gmt":"2019-12-23T18:49:02","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=164"},"modified":"2020-10-16T13:44:09","modified_gmt":"2020-10-16T13:44:09","slug":"mathcounts-exercises-3","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=164","title":{"rendered":"MATHCOUNTS Exercises – 3"},"content":{"rendered":"\n

MATHCOUNTS 2016-2017 \u2013 233<\/strong> Circle O, shown here with chords AB and BE, has secants AC and DE that intersect at X. If \\(m\\angle{ABE}=35^\\circ\\) and \\(m\\angle{AXE}=15^\\circ\\), what is the measure of \\(\\overparen{CD\\ }\\)? \"\"<\/a>
\nClick here for the solutions.<\/a><\/p>\n

\n

Solution 1<\/strong>\n$$\n\\begin{align}\n& \\because m\\angle{ABE}=\\dfrac{1}{2}\\cdot m\\overparen{AE\\ } \\\\\n& \\therefore m\\overparen{AE\\ }=2\\cdot m\\angle{ABE}=2\\cdot 35^\\circ=70^\\circ \\\\\n& \\because m\\angle{AXE}=\\dfrac{m\\overparen{AE\\ }-m\\overparen{CD\\ }}{2} \\\\\n& \\therefore m\\overparen{CD\\ }=m\\overparen{AE\\ }-2\\cdot m\\angle{AXE}=70^\\circ-2\\cdot 15^\\circ=40^\\circ \\blacksquare \\\\\n\\end{align}\n$$\n<\/p>\n

Solution 2<\/strong> Draw line \\(\\overline{OD}\\), \\(\\overline{OE}\\) and chord \\(\\overline{BF}\\) passing thru point O.\"\"<\/a><\/p>\n

\\(\\because \\triangle{AOB}\\) is isosceles
\n\\(\\therefore m\\angle{AOF}=m\\angle{OBA}+m\\angle{OAB}=2\\cdot m\\angle{OBA}\\)
\n\\(\\because \\triangle{BOE}\\) is isosceles
\n\\(\\therefore m\\angle{EOF}=m\\angle{OBE}+m\\angle{OEB}=2\\cdot m\\angle{OBE}\\)
\n\\(\\therefore m\\angle{AOE}=m\\angle{AOF}-m\\angle{EOF}=2\\cdot m\\angle{OBA}-2\\cdot m\\angle{OBE}\\)
\n\\(\\ \\ \\ \\ \\ \\ =2\\cdot(m\\angle{OBA}-m\\angle{OBE})=2\\cdot m\\angle{ABE}=2\\cdot 35^\\circ=70^\\circ\\)
\n\\(\\therefore m\\angle{OED}=m\\angle{AOE}-m\\angle{AXE}=70^\\circ-15^\\circ=55^\\circ\\)
\n\\(\\because \\triangle{DOE}\\) is isosceles
\n\\(\\therefore m\\angle{DOE}=180^\\circ-2\\cdot m\\angle{OED}=180^\\circ-2\\cdot55^\\circ=70^\\circ\\)
\n\\(\\therefore m\\overparen{CD\\ }=m\\angle{COD}=180^\\circ-m\\angle{DOE}-m\\angle{AOE}=180^\\circ-70^\\circ-70^\\circ=40^\\circ\\)
\n\\(\\blacksquare\\)\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"

MATHCOUNTS 2016-2017 \u2013 233 Circle O, shown here with chords AB and BE, has secants AC and DE that intersect at X. If \\(m\\angle{ABE}=35^\\circ\\) and \\(m\\angle{AXE}=15^\\circ\\), what is the measure of \\(\\overparen{CD\\ }\\)? Click here for the solutions. Solution 1 … Continue reading →<\/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\/164"}],"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=164"}],"version-history":[{"count":5,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/164\/revisions"}],"predecessor-version":[{"id":484,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/164\/revisions\/484"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=164"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=164"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}