exercise:70676197b0: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> Three points are chosen ''at random'' on a circle of ''unit circumference.'' What is the probability that the triangle defined by these points as vertices has three acute angles? '' Hint'': One of the angles is obtuse if and only if all three po...") |
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Three points are chosen ''at random'' on a circle of ''unit circumference.'' What is the probability that the | |||
circle of ''unit circumference.'' What is the probability that the | |||
triangle defined by these points as vertices has three acute angles? | triangle defined by these points as vertices has three acute angles? | ||
'' Hint'': One of the angles is obtuse if and only if all three points lie in the same | '' Hint'': One of the angles is obtuse if and only if all three points lie in the same | ||
semicircle. Take the circumference as the interval <math>[0,1]</math>. Take one point | semicircle. Take the circumference as the interval <math>[0,1]</math>. Take one point | ||
at 0 and the others at <math>B</math> and <math>C</math>. | at 0 and the others at <math>B</math> and <math>C</math>. | ||
Latest revision as of 23:26, 12 June 2024
Three points are chosen at random on a circle of unit circumference. What is the probability that the triangle defined by these points as vertices has three acute angles? Hint: One of the angles is obtuse if and only if all three points lie in the same semicircle. Take the circumference as the interval [math][0,1][/math]. Take one point at 0 and the others at [math]B[/math] and [math]C[/math].