A circle with center O is tangent to the coordinate axes and to the hypotenuse of the 30° - 60° - 90° triangle ABC as shown, where AB=1. To the nearest hundredth, what is the radius of the circle?

1997 AMC 12 Problem 19

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19.

A circle with center

O

is tangent to the coordinate axes and to the hypotenuse of the

30^\circ

60^\circ

90^\circ

triangle

ABC

as shown, where

AB=1.

To the nearest hundredth, what is the radius of the circle?

2.18

2.24

2.31

2.37

2.41

If the circle radius is

r

, then

O=(r,r)

in the displayed coordinates

Write the hypotenuse line of the

30^\circ

60^\circ

90^\circ

triangle and set its distance from

O

equal to

r

Take

A=(0,0),

B=(1,0),

and

C=(0,\sqrt3).

The hypotenuse is

\sqrt3x+y=\sqrt3,

and the circle center is

O=(r,r).

Tangency gives

\frac{\left|(\sqrt3+1)r-\sqrt3\right|}{2}=r.

The pictured external circle has

(\sqrt3+1)r-\sqrt3=2r,

r=\frac{\sqrt3}{\sqrt3-1}

=\frac{3+\sqrt3}{2}\approx2.37.

Thus D is correct.