Solid Mensuration

A triangle has a base of 16 inches and an altitude of 8 inches. Find the dimensions of the largest rectangle that can be inscribed in the triangle if the base of the rectangle coincides with the base of the triangle.

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asked by Jean
  1. I suspect it will be 4x8, but let's see what the calculations show.

    If triangle ABC has vertices at (0,0),(16,0) and (h,8) then the sloping sides are lines with the equations

    y = 8/h x
    y = 8(x-16)/(h-16)

    Now, suppose the inscribed rectangle has height k. The the line y=k intersects the sides where x is

    hk/8 and k(h-16)/8 + 16

    The length of the rectangle is thus 16-2k

    The area of the rectangle is

    a = k(16-2k) = 16k-2k^2
    The vertex of this parabola is where k=4.

    Thus the inscribed rectangle of maximal area is half the height of the triangle, making its base half the base of the triangle. The rectangle is thus 4x8.

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    posted by Steve

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