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# Circle Help!

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A quarter-circle is drawn inside a square, as shown below. A point on the quarter-circle has distances of 1 and 8 from two sides of the square.
Find the side length of the square.

https://latex.artofproblemsolving.com/miscpdf/tledzvvo.pdf?t=1574736375974

Sorry I forgot the picture

Nov 26, 2019
edited by MathCuber  Nov 26, 2019

#2
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Set the side lengths equal to x. That means that the small rectangle has a width 1/x, and a length of 8/x.

The area of the rectangle in terms of x is therefore: 1/x * 8/x = 8/x2

And you can calculate the area of the rectangle as 1 * 8 = 8.

So you have the equation of 8 = 8/x2

I will leave you to solve that, it should take less than 30 seconds.

This is CalcUser not logged in, thanksgiving break has granted me a little more time on this website.

Nov 26, 2019
#3
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wait this is a bogus solution

Guest Nov 26, 2019
#4
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Let the radius  of the circle  and the side of the square  =  R

If we let the bottom right of the square  =  (0, 0)  then we can find P  as  ( -R + 8, R - 1)

So.....we have that

(-R + 8)^2  + (R - 1)^2   =  R^2       simplify

R^2 - 16R + 64  +R^2 - 2R + 1  = R^2

R^2 - 18R + 65  =  0      factor

(R - 13) ( R - 5)  =  0

Since R  must be greater than 8.......the radius must be  13  = side of the square

Here's a pic :    Nov 26, 2019
#5
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Thank you so much! This makes a lot more sense!

MathCuber  Nov 26, 2019