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.

Guest Mar 23, 2020

#1**+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)

CPhill Mar 23, 2020

#2**+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.

geno3141 Mar 23, 2020