AMC10 2007 B

AMC10 2007 B · Q18

AMC10 2007 B · Q18. It mainly tests Quadratic equations, Circle theorems.

A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?

如图，一个半径为1的圆被4个半径为$r$的圆包围。求$r$？

(A) $\sqrt{2}$ $\sqrt{2}$

(B) $1 + \sqrt{2}$ $1 + \sqrt{2}$

(D) 3 3

(E) $2 + \sqrt{2}$ $2 + \sqrt{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Construct the square ABCD by connecting the centers of the large circles, as shown, and consider the isosceles right $\triangle BAD$. Since AB = AD = 2r and BD = 2 + 2r, we have 2(2r)$^2$ = (2 + 2r)$^2$. So 1 + 2r + r$^2$ = 2r$^2$, and r$^2$ − 2r − 1 = 0. Applying the quadratic formula gives r = 1 + $\sqrt{2}$.

连接大圆圆心构成正方形ABCD，如图所示，考虑等腰直角$\triangle BAD$。因为$AB = AD = 2r$，$BD = 2 + 2r$，有$2(2r)^2 = (2 + 2r)^2$。即$1 + 2r + r^2 = 2r^2$，$r^2 − 2r − 1 = 0$。套用二次公式得$r = 1 + \sqrt{2}$。

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