Average area of triangle formed 1 8 that of square.A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is one eighth of the area of the square.

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Average area of triangle formed             1      8       that of square.A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is one eighth of the area of the square.

procjenomuj · Accepted Answer

Step 1In compound probability product of two different probabilities   A  r  e  a  =      1    2    x  ⋅  y  ;           p          A      r      e      a        =      1    2          p    x          p    y  Step 2Here       p    x    =      p    y    =      1    2   for the full range of side length (0,1) consideredWe can directly compare the area of gray triangle area to that of the square                     1        2            ⋅      (              1        2                    )        2                    1      2        =      1    8

seguitzla · Accepted Answer

Step 1Inside your square, draw the quadrilateral with vertices (p,0), (0,q), (p,1), (1,q). The triangles in the corners of the square have total area 1/2. Proof: Draw the dashed lines below. Each triangle is half the area of the rectangle containing it, and those rectangles add up to area 1.Step 2Thus, the average area of these 4 triangles is 1/8. By the symmetry of the problem, therefore, the expected area of each of the triangles is 1/8.

A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is one eighth of the area of the square. Average area of triangle formed 1/8 that of square

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