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A quarter-circle with radius  is drawn. A circle is drawn inside the sector, which is tangent to the sides of the sector, as shown. Find the radius of the inscribed circle.

 Mar 23, 2020
 #1
avatar+111330 
+1

Call R  the radius of the  larger quarter-circle and  r  be  the  radius  of the smaller circle

 

Let P  be  the  center of the  smaller circle

 

And we can  show that the distance  from O to P =  √2r

 

Then

 

R =  r  + √2 r

 

R = r ( 1 + √2)

 

So

 

r =    R  / ( 1 + √2)

 

 

cool cool cool

 Mar 23, 2020
 #2
avatar+21017 
+1

Let  P  be the point of tangency of the small circle with line segment AO.

Let  Q  be the point of tangency of the small circle with line segment BO.

Let  R be the center of the small circle.

 

OPRQ is a square.

Let the distance from O to P be x.

Then the radius of the small circle is also x.

Since OPRQ is a square with each side x, the distance from O to R is  x·sqrt(x).

Then the radius of the large circle (with center O and radius OA) is  x + x·sqrt(x).

 

Now, if you know the radius of the large circle, you can find the radius of the small circle.

 Mar 23, 2020

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