+1725 A circle lies inside a quarter-circle, as shown below. The circle is tangent to side $\overline{AO}$ and arc $AB$ and side $\overline{BO}$. If the radius of the quarter-circle si 2, then find the radius of the circle.
+129771 Let X be the point of tangency to the edges of both circles
Let the other two tangent points at the intersection points of the radius of the quarter-circle and the edge of the small circle be Y and Z
Let the center of the small circle = M
Connect MY , MZ and MO
Triangles MYO and MZO form congruent 45 - 45 - 90 right triangles
Let the radius of the small circle = r
MZ = r MO = (sqrt2) * r
Connect OX
OX = MO + MX
OX = 2 MO = sqrt2 * r MX = r
So
MO + MX = OX
sqrt (2)r + r = 2
r ( sqrt 2 + 1) = 2
r = 2 / (sqrt (2) + 1) ≈ .828
