{"id":4303,"date":"2024-07-22T15:40:00","date_gmt":"2024-07-22T19:40:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4303"},"modified":"2024-10-25T01:58:14","modified_gmt":"2024-10-25T05:58:14","slug":"geometry-probability-1","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4303","title":{"rendered":"Geometry Probability &#8211; 1"},"content":{"rendered":"\n<p>3 random points are chosen independently on a circle. What is probability that they form an acute triangle? Click <a onclick=\"toggle_visibility('geometry-probability-1');\">here<\/a> for the solution.\n<\/p>\n\n\n\n<div id=\"geometry-probability-1\" style=\"display:none\">\n\n\n\n<p><strong>Solution<\/strong>: This problem is equivalent to the following:<\/p>\n\n\n\n<p>Two cuts are randomly made on a line of length $180$, dividing the line into three segments. The lengths of the three segments are taken as the value in degrees of $\\angle{A}$, $\\angle{B}$, and $\\angle{C}$ of $\\triangle{ABC}$. What is the probability that $\\triangle{ABC}$ is acute?<\/p>\n\n\n\n<p>$$\\angle{A}+\\angle{B}+\\angle{C}=180, \\angle{A}&lt;90, \\angle{B}&lt;90, \\angle{C}&lt;90$$<\/p>\n\n\n\n<p>Let $x=\\dfrac{\\angle{A}}{180}$, $y=\\dfrac{\\angle{B}}{180}$, $z=\\dfrac{\\angle{C}}{180}$, we have:<\/p>\n\n\n\n<p>$$x+y+z=1, x&lt;\\dfrac{1}{2}, y&lt;\\dfrac{1}{2}, z&lt;\\dfrac{1}{2}$$<\/p>\n\n\n\n<p>Therefore $$x+y&gt;\\dfrac{1}{2}, x+y&lt;1, x&lt;\\dfrac{1}{2}, y&lt;\\dfrac{1}{2} \\tag{1}$$<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-24-at-3.24.39\u202fPM-1024x1024.png\" alt=\"\" class=\"wp-image-4309\" width=\"320\" height=\"320\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-24-at-3.24.39\u202fPM-1024x1024.png 1024w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-24-at-3.24.39\u202fPM-300x300.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-24-at-3.24.39\u202fPM-150x150.png 150w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-24-at-3.24.39\u202fPM-768x768.png 768w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-24-at-3.24.39\u202fPM.png 1280w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><\/figure>\n\n\n\n<p>In the above diagram, $\\triangle{ODE}$ represents all possible $x$ and $y$ values. And $\\triangle{ABC}$ represents all $x$ and $y$ values satisfying $(1)$. Therefore, the probability to the original question is $\\dfrac{1}{4}$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>3 random points are chosen independently on a circle. What is probability that they form an acute triangle? Click here for the solution. Solution: This problem is equivalent to the following: Two cuts are randomly made on a line of &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=4303\">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],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4303"}],"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=4303"}],"version-history":[{"count":10,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4303\/revisions"}],"predecessor-version":[{"id":4553,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4303\/revisions\/4553"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4303"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4303"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}