2017 AMC 12A 第 16 题

Question

在下图中，以

A

和

B

为圆心、半径分别为

2

和

1

的半圆，画在一个以

\overline{JK}

为直径的半圆内部，并与它共用底边。两个较小的半圆彼此外切，并且都与最大半圆内切。以

P

为圆心的圆与两个较小半圆外切，并与最大半圆内切。以

P

为圆心的圆的半径是多少？

In the figure below, semicircles with centers at

A

and

B

and with radii

2

and

1,

respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter

\overline{JK}.

The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at

P

is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at

P?

\dfrac{3}{4}

\dfrac{6}{7}

\dfrac{1}{2}\sqrt3

\dfrac{5}{8}\sqrt2

\dfrac{11}{12}

答案：B

知识点：相切圆勾股定理坐标几何

难度评级：1840

小提示：

设

C

为大半圆圆心；则

PA=2+r

，

PB=1+r

，

PC=3-r

。

Let

C

be the center of the large semicircle; then

PA=2+r,

PB=1+r,

PC=3-r

大提示：

从

P

向

\overline{JK}

作垂线，并令三个关于高度平方的表达式相等

Drop a perpendicular from

P

to

\overline{JK}

and set the three expressions for its squared height equal

解答：

大半圆半径为

3

，圆心

C

是

\overline{JK}

的中点。把

J

放在原点，则沿底边有

A=2

、

B=5

、

C=3

、

K=6

。设

P

处圆的半径为

r

。

由相切可知

PA=2+r

，

PB=1+r

，且

PC=3-r

。从

P

向底边作垂线，垂足的水平位置为

3+x

，高度为

h

，由勾股定理得

\begin{aligned} h^2 &=(2+r)^2-(1+x)^2 \\ &=(3-r)^2-x^2 \\ &=(1+r)^2-(2-x)^2 \end{aligned}\text{。}

令第一个表达式与中间那个相等，得到

5r-x=3

；令最后一个表达式与中间那个相等，得到

2r+x=3

。两式相加得

7r=6

，所以

r=\dfrac{6}{7}

。

所以正确答案是 B。

The large semicircle has radius

3

and center

C,

the midpoint of

\overline{JK}.

Placing

J

at the origin,

A=2,

B=5,

C=3,

K=6

along the base. Let

r

be the radius of the circle at

P.

By tangency,

PA=2+r,

PB=1+r,

and

PC=3-r.

Dropping a perpendicular from

P

to the base at horizontal position

3+x

with height

h,

the Pythagorean theorem gives

\begin{aligned} h^2 &=(2+r)^2-(1+x)^2 \\ &=(3-r)^2-x^2 \\ &=(1+r)^2-(2-x)^2. \end{aligned}

Equating the first expression with the middle one gives

5r-x=3,

while equating the last expression with the middle one gives

2r+x=3.

Adding yields

7r=6,

so

r=\dfrac{6}{7}.

Thus, the correct answer is B.

2017 AMC 12A 第 16 题

答案：B

难度评级：1840

小提示：

大提示：

解答：

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