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Four unit circles surround a smaller circle.  Find the radius of the smaller circle.

 

 Dec 16, 2019
 #1
avatar+111321 
+1

Find the centers of each of the 4  larger circles and connect them

 

This will form a square of 2 units on each side

 

And the distance from the center  of the circle at the top left to the  center of the circle on the bottom right  =

2√2  units

 

So......the diameter of the small circle =  2√2 - 2   = 2 [ √2 - 1]

 

And its radius is 1/2 of this  = [ √2 - 1 ]  units 

 

 

cool cool cool

 Dec 16, 2019
edited by CPhill  Dec 16, 2019
 #2
avatar+993 
+1

Hi wiseowl!

 

Connect the centers like in the following diagram: https://imgur.com/WoRzy2n

 

This forms a right triangle with equal legs of 2 units. 

 

The hypotenuse is equal to the diameter of the smaller circle plus 2. 

 

We can right the equation: \(2^2+ 2^2=(2+2x)^2\\ 8=4+8x+4x^2\\ x^2+2x-1=0\) 

 

Solving for x, we get x = √2 - 1

 

I hope this helped,

 

Gavin

 Dec 16, 2019

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