AMC12 2011 B

AMC12 2011 B · Q6

AMC12 2011 B · Q6. It mainly tests Angle chasing, Circle theorems.

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 $\angle{BAC}$?

从一点 $A$ 引出两条圆的切线。切点 $B$ 和 $C$ 将圆分成弧，其长度比为 $2:3$。$\angle{BAC}$ 的度量是多少度？

(A) 24 24

(B) 30 30

(D) 48 48

(E) 60 60

Answer

Correct choice: (C)

正确答案：（C）

Solution

In order to solve this problem, use of the tangent-tangent intersection theorem (Angle of intersection between two tangents dividing a circle into arc length A and arc length B = 1/2 (Arc A° - Arc B°). In order to utilize this theorem, the degree measures of the arcs must be found. First, set A (Arc length A) equal to 3d, and B (Arc length B) equal to 2d. Setting 3d+2d = 360° will find d = 72°, and so therefore Arc length A in degrees will equal 216° and arc length B will equal 144°. Finally, simply plug the two arc lengths into the tangent-tangent intersection theorem, and the answer: 1/2 (216°-144°) = 1/2 (72°) $=\boxed{36 \textbf{(C)}}.$

为了解此题，使用切线-切线交角定理（两条切线的夹角等于所截两弧度数之差的一半，即 $\frac12(\text{弧}A^\circ-\text{弧}B^\circ)$）。为使用该定理，需要先求两段弧的度数。设弧长比中较长的弧为 $3d$，较短的弧为 $2d$。由 $3d+2d=360^\circ$ 得 $d=72^\circ$，因此较长弧的度数为 $216^\circ$，较短弧的度数为 $144^\circ$。代入切线-切线交角定理： $\frac12(216^\circ-144^\circ)=\frac12(72^\circ) $ $=\boxed{36 \textbf{(C)}}.$

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