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

MathCuber Nov 26, 2019

#2**0 **

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/x^{2}

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

So you have the equation of 8 = 8/x^{2}

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.

Guest Nov 26, 2019

#4**+2 **

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 :

CPhill Nov 26, 2019