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.
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.
Details:
A = (0,0)
B = (0,1)
C = (1,1)
D = (1,0)
E = (1u, 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 20191118 13:58:24 