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Square and Rectangle (Posted on 2019-07-04) Difficulty: 3 of 5
ABCD is a square with point E on BC. DEFG is a rectangle with point A on FG.

Let x=CE. Express area(ABCD)/area(DEFG) in terms of x.

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (1 votes)

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Solution Algebra confirms Comment 3 of 3 |
I used the letter 'u' instead of 'x' for CE because I used an xy coordinate system making ABCD a unit square with A at the origin.  Figured the equation for the line which includes segment DE.  Figured the equation of a perpendicular line through the origin and the intersection point.  From there determined the distance from the origin to the line segment:  this was sqrt(1+u^2) / (1+u^2).
And figured the length of DE:  sqrt(1+u^2).

So, the area of the square was 1, and
the area of the rectangle was sqrt(1+u^2)* sqrt(1+u^2) / (1+u^2).
Also 1.

So the ratio is 1 and independent of the length of CE.

A = (0,0)
B = (0,1)
C = (1,1)
D = (1,0)
E = (1-u, 1)
line DE:  y = -(1/u)x + 1/u
line perpendicular to DE and through origin:  y = ux
point of intersection:  (1/(1+u^2), u/(1+u^2)) from which
distance from A to line DE: sqrt(1+u^2) / (1+u^2)

  Posted by Larry on 2019-11-18 13:58:24
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