题目

Problem

The diagram shows a sector of a circle, centre $O$ , where $OB=OC=15$ cm. The size of angle $BOC$ is $\dfrac{2}{5}\pi$ radians. Points $A$ and $D$ on the lines $OB$ and $OC$ respectively are joined by an arc $AD$ of a circle with centre $O$ . The shaded region is bounded by the arcs $AD$ and $BC$ and by the straight lines $AB$ and $DC$ . It is given that the area of the shaded region is $\dfrac{209}{5}\pi\text{ cm}^2$ .

Find the perimeter of the shaded region. Give your answer in terms of $\pi$ .

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题目中文翻译

图中显示一个圆心为 $O$ 的扇形，其中 $OB=OC=15$ cm。角 $BOC$ 的大小为 $\dfrac{2}{5}\pi$ 弧度。点 $A$ 和点 $D$ 分别在直线 $OB$ 和 $OC$ 上，并由一个以 $O$ 为圆心的圆弧 $AD$ 连接。阴影区域由圆弧 $AD$ 、圆弧 $BC$ 以及直线 $AB$ 、 $DC$ 围成。已知阴影区域的面积为 $\dfrac{209}{5}\pi\text{ cm}^2$ 。

求阴影区域的周长。答案用 $\pi$ 表示。

解答

解法一

思路

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阴影面积是两个同角扇形的面积差。求出内半径后，周长由两段圆弧和两段径向直边组成。

答题过程

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Let the inner radius be $r$ . Since the common angle is $2\pi/5$ ,

\frac12\cdot\frac{2\pi}5(15^2-r^2)=\frac{209\pi}5.

It follows that $r^2=225-209=16$ , so $r=4$ . The two arcs have total length $(15+4)\cdot2\pi/5$ , and the two straight edges have total length $2(15-4)$ . The perimeter is therefore

22+\frac{38\pi}5\text{ cm}.

CIE 9709 2024 November Paper 11 Q3

题目

解答

解法一

思路

答题过程