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#25 Grid-in

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Do those marks through the figure indicate sides that are the same? I forget stupid notations like that.

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Yep. So x=CD.


I can't see why x can't have a wide range of values, i.e. 0<x<9.


[edited stupid inequalities]

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Since both sides of the triangle on the bottom left = x and theta=90


each other angle is 45 degrees.


So the angle in the middle is 180-90-45 = 45


If we draw a horizontal line from the point above X (O) to line AC, and say the intersection is point P,


the hypoteneuse is still sqrt(2x^2), each side is x


Now, AC = AP + PC


and we know PC = x


We now have a triangle APO, with a right angle and side x


sin(:lctheta:) = AP / x

AP = x*sin(:lctheta:)


AC = x+x*sin(theta)


So the area is x*(x+xsin(:lctheta:))


Now :lctheta: is some angle between 0 and 90 (because we already have 2 45 degree angles along that line)

Between 0 and 90, which means sin(:lctheta:) is between 0 and 1.


Solving for 8 and 18, we find that if we have the case of the minumum area of 8,

x=2sqrt(2)/sqrt(sin(:lctheta:) + 1)

and that if the maximum area of 18 is the case,

x=3sqrt(2)/sqrt(sin(:lctheta:) + 1)


Playing around with those numbers, you eventually get that x is going to be between 2 and 4.3.


In any case, I personally think that question would be thrown out because of how time consuming the correct answer set is.

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