Small Circle

A circle is inscribed in a 60^\circ sector of a circle of radius 1 (as shown). What is the radius of the small circle?

Source: NCTM Mathematics Teacher, February 2006

SOLUTION

\overline{AC} is the angle bisector of \angle A because C is equidistant from \overline{AB} and \overline{AD}. Triangle ABC is a 30^\circ\!\textrm{-}60^\circ\!\textrm{-}90^\circ triangle with leg BC=x and hypotenuse AC=2x.
AC+CF=1
2x+x=1
x=1/3
The radius of the small circle equals 1/3.

Answer: 1/3

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About mvtrinh

Retired high school math teacher.
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