{"id":4847,"date":"2025-02-02T14:10:00","date_gmt":"2025-02-02T18:10:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4847"},"modified":"2025-03-01T12:03:12","modified_gmt":"2025-03-01T16:03:12","slug":"geometry-probability-4","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4847","title":{"rendered":"Geometry Probability – 4"},"content":{"rendered":"\n

Let $D$ is an interior point inside equilateral $\\triangle{ABC}$. Find the probability that the line segments of $AD$, $BD$, and $CD$ are the side of: (1) a triangle, (2) a right triangle, (3) an obtuse triangle, and (4) an acute triangle. Click here<\/a> for answers.<\/p>\n\n\n\n

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(1) Rotating $\\triangle{ACD}$ around $C$ counter-clockwise $60^\\circ$ so that $D$ is moved to point $D’$, and $A$ to $B$, we have $\\triangle{DCD’}$ as an equilateral triangle, and $\\triangle{ABD}\\cong\\triangle{ABD’}$. Therefore, $AD=BD’$, $CD=DD’$. Thus $AD$, $BD$, and $CD$ form $\\triangle{DBD’}$. Therefore, the probability is $1$.<\/p>\n\n\n\n

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(2) If $\\angle{BDC}=150^\\circ$, $\\angle{BDD’}=\\angle{BDC}-\\angle{CDD’}=150^\\circ-60^\\circ=90^\\circ$. Therefore, if $D$ is on the red arcs in the following diagram, $AD$, $BD$, and $CD$ will form a right triangle. Therefore, the probability is $0$.<\/p>\n\n\n\n

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(3) If $D$ is inside the regions formed by arcs and the sides of $\\triangle{ABC}$, then $AD$, $BD$, and $CD$ will form an obtuse triangle. The probability is $$\\dfrac{3\\cdot\\Big(\\dfrac{\\pi}{6}-\\dfrac{\\sqrt{3}}{4}\\Big)}{\\dfrac{\\sqrt{3}}{4}}=\\dfrac{2\\pi\\sqrt{3}}{3}-3\\approx 0.6276$$<\/p>\n\n\n\n

(4) The probability is $$1-\\Big(\\dfrac{2\\pi\\sqrt{3}}{3}-3\\Big)=4-\\dfrac{2\\pi\\sqrt{3}}{3}\\approx 0.3724$$<\/p>\n\n\n\n<\/div>\n\n\n\n

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Let $D$ is an interior point inside equilateral $\\triangle{ABC}$. Find the probability that the line segments of $AD$, $BD$, and $CD$ are the side of: (1) a triangle, (2) a right triangle, (3) an obtuse triangle, and (4) an acute … 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":[9,8],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4847"}],"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=4847"}],"version-history":[{"count":16,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4847\/revisions"}],"predecessor-version":[{"id":4944,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4847\/revisions\/4944"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4847"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4847"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}