Determine Area (Posted on 2010-03-27)

Let ABHF be a trapezoid such that 

   1) sides AB and FH are parallel,
   2) angle BAF is 90 degrees, and
   3) side FH is larger than side AB.

Let G be a point on side FH such that line segment BG 
is perpendicular to FH. Let line segment CDE be parallel 
to AB with points C, D, and E on line segments AF, BG, 
and BH respectively.

Let x = [ABDC], y = [CDGF], z = [BDE], and w = [DEHG].

Determine w in terms of x, y, and z.

Note: [PQ...UV] denotes the area of polygon PQ...UV.

	Submitted by Bractals
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Solution:	(Hide)
Ratio of rectangle areas: x |BD||CD| |BD| ----- = ---------- = ------ (1) x+y |BG||CD| |BG| Ratio of similar triangle areas: z |BD| ----- = [------]2 (2) z+w |BG| Combining (1) and (2), z x ----- = [-----]2 z+w x+y Solving for w, yz(2x+y) w = ---------- x2

	Subject	Author	Date
	Puzzle Answer	K Sengupta	2023-12-18 01:00:01
	solution	Charlie	2010-03-27 16:00:37