Radius of Tangent Circle

A circular arc centered at one vertex of a square of side length 4 passes through two other vertices. A small circle is tangent to the large circle and to two sides of the square. What is the radius of the small circle?

Source: NCTM Mathematics Teacher 2006

SOLUTION

Triangle ABC is a 45^\circ\!\textrm{-}45^\circ\!\textrm{-}90^\circ triangle of side length r
AB=r\sqrt 2
AD is the diagonal of a square of side length 4
AD=4\sqrt 2
AD=AB+BD
4\sqrt 2=r\sqrt 2+(r+4)
4\sqrt 2-4=r\sqrt 2+r
4(\sqrt 2-1)=r(\sqrt 2+1)
r=\dfrac{4(\sqrt 2-1)}{\sqrt 2+1}
Rationalize
r=\dfrac{4(\sqrt 2-1)^2}{(\sqrt 2+1)(\sqrt 2-1)}
=\dfrac{4(2-2\sqrt 2+1)}{2-1}
=4(3-2\sqrt 2)
=12-8\sqrt 2 unit

Answer: 12-8\sqrt 2 unit

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About mvtrinh

Retired high school math teacher.
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