{"id":4393,"date":"2024-09-18T02:09:00","date_gmt":"2024-09-18T06:09:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4393"},"modified":"2024-10-25T15:14:21","modified_gmt":"2024-10-25T19:14:21","slug":"geometry-probability-3","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4393","title":{"rendered":"Geometry Probability – 3"},"content":{"rendered":"\n

Two points are randomly and uniformly selected from the interior of a circle. The center of the circle and the two points joined together form a triangle. What is the probability that the triangle is acute? Click here<\/a> for the solution.<\/p>\n\n\n\n

\n\n\n\n

Solution: <\/strong>First pick up the first point A randomly and uniformly from the interior of the unit circle centered at $O$ and orientate $OA$ be on positive side of the $x$-axis, with the distance between $O$ and $A$ as $t$:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Draw another circle with $OA$ as its diameter. Draw a chord $DE \\perp OA$ and link $OD$ and $OE$. <\/p>\n\n\n\n

If point $B$ is selected from the left side of the circle, $\\triangle{OAB}$ would be obtuse. Therefore, in order for the another point $B$ randomly and uniformly selected from the interior of the circle so that $\\triangle{OAB}$ is acute, point B must be on the right half of the unit circle.<\/p>\n\n\n\n

Additionally, point $B$ must be not inside the smaller circle with its diameter as $OA$, and not on the right side of line $DE$.<\/p>\n\n\n\n

Therefore, the probability $\\triangle{OAB}$ is obtuse when $B$ is on the right half of the unit circle for $t$ would be: <\/p>\n\n\n\n

$$f(t)=\\dfrac{area\\ of\\ the\\ smaller\\ circle\\ + \\ area\\ of\\ unit\\ circle\\ on\\ the\\ rigit\\ side\\ of \\ line\\ DE}{area\\ of\\ the\\ semi\\ unit\\ circle}$$ $$=\\dfrac{\\dfrac{1}{4}\\pi t^2 + cos^{-1}(t)-t\\sqrt{1-t^2}}{\\dfrac{1}{2}\\pi} \\tag{1}$$<\/p>\n\n\n\n

Therefore the probability of $\\triangle{OAB}$ is obtuse on the right half of the unit circle would be:<\/p>\n\n\n\n

$$P=\\int_{0}^{1}f(\\sqrt{x})dx \\tag{2}$$<\/p>\n\n\n\n

$\\sqrt{x}$ is used to ensure that point $A$ is uniformly distributed across whole unit circle area. The square root function compensates for the quadratic increase in area, spreading the points evenly.<\/p>\n\n\n\n

Simplifying $(2)$ we have:<\/p>\n\n\n\n

$$P=\\dfrac{1}{2}\\int_{0}^{1} x dx + \\dfrac{2}{\\pi}\\int_{0}^{1}cos^{-1}(\\sqrt{x}) dx -\\dfrac{2}{\\pi}\\int_{0}^{1}\\sqrt{x(1-x)} dx$$<\/p>\n\n\n\n

$$=\\dfrac{1}{2}\\cdot \\dfrac{1}{2}+\\dfrac{2}{\\pi}\\cdot \\dfrac{\\pi}{4}-\\dfrac{2}{\\pi}\\cdot \\dfrac{\\pi}{8}=\\dfrac{1}{2}$$<\/p>\n\n\n\n

Therefore the probability of $\\triangle{OAB}$ is acute on the right half of the unit circle is $1-P=\\dfrac{1}{2}$.<\/p>\n\n\n\n

Since the probability for $B$ to be in the right half of the unit circle is $\\dfrac{1}{2}$, the overall probability is $\\dfrac{1}{2}\\cdot \\dfrac{1}{2}=\\boxed{\\dfrac{1}{4}}$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Two points are randomly and uniformly selected from the interior of a circle. The center of the circle and the two points joined together form a triangle. What is the probability that the triangle is acute? Click here for the … Continue reading →<\/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":[13,17,9,8],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4393"}],"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=4393"}],"version-history":[{"count":23,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4393\/revisions"}],"predecessor-version":[{"id":4544,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4393\/revisions\/4544"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}