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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
 #1
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as shown where?

 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 :

 

 

 

cool cool cool

 Nov 26, 2019
 #5
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Thank you so much! This makes a lot more sense!

MathCuber  Nov 26, 2019

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