2011 AMC 12B Problem 6

Below is the professionally curated solution for Problem 6 of the 2011 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 12B solutions, or check the answer key.

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Concepts:tangent line arc angle chasing

Difficulty rating: 1240

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 $\angle BAC?$

$24$

$30$

$36$

$48$

$60$

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 $ABOC,$ $\angle BAC=360^\circ-144^\circ-90^\circ-90^\circ=36^\circ.$

Thus, the correct answer is C.

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