Length of one side for maximum area of trapezoid (solution by Geometry)

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Problem
BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.

Trapezoid with one side tangent to the circle

Solution

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As described by Alexander Bogomolny of cut-the-knot.org, for maximum area of trapezoid, the point of tangency H should be at the midline of AB and DC, thus H is the midpoint of BC.

From the figure:
$z^2 + 5^2 = 15^2$

$z = 10\sqrt{2}$

For the angle theta:
$\cos \theta = \dfrac{z}{15} = \dfrac{20}{BC}$

$BC = \dfrac{300}{z} = \dfrac{300}{10\sqrt{2}}$

$BC = 15\sqrt{2} \, \text{ cm} = 21.21 \, \text{ cm}$ answer

A solution to the same problem has been carried out with the aid of Calculus. You can see how ugly it is without the above analogy of Alexander Bogomolny. See this link to to see the solution by maxima and minima: Solution by Calculus

‹ Trapezoidal Strip of Land from a Triangular Lot up The Cyclic Quadrilateral ›

Submitted by Romel Verterra on June 3, 2012 - 5:51pm

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Length of one side for maximum area of trapezoid (solution by Geometry)

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Length of one side for maximum area of trapezoid (solution by Geometry)

Recent Updates

Plane Geometry

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