2018 AMC 12B 第 4 题

Question

一个圆有一条长度为

10

的弦，且圆心到这条弦的距离为

5

。这个圆的面积是多少？

A circle has a chord of length

10,

and the distance from the center of the circle to the chord is

5.

What is the area of the circle?

25\pi

50\pi

75\pi

100\pi

125\pi

半径、半条弦和圆心到弦的距离构成直角三角形。

The radius, half the chord, and the distance to the chord form a right triangle

两条直角边为

5

和

5

，所以半径满足

r^2=5^2+5^2

。

With legs

5

and

5,

the radius satisfies

r^2=5^2+5^2

从圆心向弦作垂线会平分弦，形成直角三角形，其两条直角边分别为

5

（半弦长）和

5

（圆心到弦距离），斜边为

r

。

因此

r^2=5^2+5^2=50

，圆面积为

\pi r^2=50\pi

。

所以正确答案是 B。

Dropping a perpendicular from the center to the chord bisects it, forming a right triangle with legs

5

(half the chord) and

5

(the distance), and hypotenuse

r.

Then

r^2=5^2+5^2=50,

so the area is

\pi r^2=50\pi.

Thus, the correct answer is B.

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