2011 AMC 12B Problem 6

Question

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?

24

30

36

48

60

The two arcs split

360^\circ

in the ratio

2:3

A tangent meets the radius at its point of contact at

90^\circ

; use quadrilateral

ABOC

Let

O

be the center. The arcs measure

2x

and

3x

with

2x+3x=360^\circ,

so

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.

Problem 6 in other years