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This is a challenging problem from the 1986 American Invitational Mathematics Exam (AIME).<\/p>\nThis is a challenging problem from the 1986 American Invitational Mathematics Exam (AIME).<\/p>\n
\u201cLet triangle ABC<\/em> be a right triangle in the xy<\/em>-plane with a right angle at C<\/em>. Given that the length of the hypotenuse AB<\/em> is 60, and that the medians through A<\/em> and B<\/em> lie along the lines y = x<\/em> + 3 and y<\/em> = 2x<\/em> + 4 respectively, find the area of triangle ABC<\/em>.\u201d<\/p>\n I have included a sketch to indicate that the sides of the right triangle are not parallel to the Cartesian coordinate axes.\u00a0<\/p>\n The AIME (American Invitational Mathematics Examination<\/a>) is an intermediate examination between the American Mathematics Competitions<\/a> AMC 10 or AMC 12 and the USAMO (United States of America Mathematical Olympiad). All students who took the AMC 12 (high school 12th<\/sup> grade) and achieved a score of 100 or more out of a possible 150 or were in the top 5% are invited to take the AIME. All students who took the AMC 10 (high school 10th<\/sup> grade and below) and had a score of 120 or more out of a possible 150, or were in the top 2.5% also qualify for the AIME.<\/p>\n
![Answer<\/a>.<\/p>\nSee the Challenging Triangle Problem<\/a> for solutions.<\/p>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"This is a challenging problem from the 1986 American Invitational Mathematics Exam (AIME). \u201cLet triangle ABC be a right triangle in the xy-plane with a right angle at C. Given that the length of the hypotenuse AB is 60, and that the medians through A and B lie along the lines y = x + […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[238,40],"class_list":["post-2608","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-aime","tag-analytic-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2608","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=2608"}],"version-history":[{"count":3,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2608\/revisions"}],"predecessor-version":[{"id":3636,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2608\/revisions\/3636"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=2608"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=2608"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=2608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}](\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Challenging-Triangle-Answer.jpg\"){kind=link}