图中所示为一个圆心为 $A$、半径为 $5\text{ cm}$ 的圆，以及一个圆心为 $B$、半径为 $8\text{ cm}$ 的圆。两圆在点 $C$ 处相切，$A$、$C$、$B$ 在同一直线上。较小的圆在点 $D$ 处的切线与较大的圆交于点 $E$，并经过点 $B$。

(a) 求阴影区域的周长。

(b) 求阴影区域的面积。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

半径垂直于切线，故 $ADB$ 是直角三角形。用它求直边和两个扇形的圆心角。

The centres are $AB=5+8=13$ cm apart. Since $AD=5$ is perpendicular to the tangent, $DB=\sqrt{13^2-5^2}=12$ cm. The point $E$ lies between $D$ and $B$, so $DE=12-8=4$ cm. The sector angles in radians are

at $A$ and $B$ respectively. The perimeter is $5\alpha+8\beta+4=13.0$ cm to three significant figures.

阴影位于直角三角形内。用三角形面积减去两个扇形面积。

The area of triangle $ADB$ is $\tfrac12(5)(12)=30$ square centimetres. Using the unrounded angles from (a), the shaded area is

which is $2.67\,\mathrm{cm}^2$ to three significant figures.