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Thread: rectangle

  1. #1
    Senior Member Sampras's Avatar
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    rectangle

    Consider the unit circle C: $\displaystyle x^2+y^2 = 1 $. Suppose 2 points are chosen randomly: (i) $\displaystyle p $ is chosen from the circumference and (ii) $\displaystyle q $ is chosen from the interior (chosen independently). Let $\displaystyle R $ be the rectangle with diagonal $\displaystyle pq $. What is the probability that no point of R lies outside of C?

    Wouldn't a rectangle with diagonal $\displaystyle pq $ always lie inside the circle?
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  2. #2
    MHF Contributor
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    The Diagonal pq will always be in or on the circle. Not so the other diagonal.

    Easy demonstration:

    $\displaystyle p = <0, 1>$

    $\displaystyle q = <1-\epsilon , 0>$

    Where $\displaystyle 0 < \epsilon << 1$

    p is the North West Vertex
    q is the South East Vertex

    Where is the North East vertex?
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