2011 AMC 12B Problem 6

Problem 6 of 25EasierGeometry

Two tangents to a circle are drawn from a point

A.

The points of contact

B

and

C

divide the circle into arcs with lengths in the ratio

2:3.

What is the degree measure of

\angle BAC?

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Solution

Let

O

be the center. The arcs measure

2x

and

3x

with

2x+3x=360^\circ,

x=72^\circ

and the minor arc

BC

gives central angle

\angle BOC=144^\circ.

The radii to

B

and

C

are perpendicular to the tangents, so

\angle ABO=\angle ACO=90^\circ.

In quadrilateral

ABOC,

\begin{gathered} \angle BAC=360^\circ-144^\circ-90^\circ \\ {}-90^\circ=36^\circ. \end{gathered}

Thus, the correct answer is C.

Tagged: tangent line · arc · angle chasing