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 $\angle BAC?$

$24$

$30$

$36$

$48$

$60$

Answer: C

Concepts:tangent line arc angle chasing

Difficulty rating: 1240

Solution:

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.

2011 AMC 12B Problem 6

Answer: C

Difficulty rating: 1240

Solution:

Problem 6 in Other Years