AMC10 2002 A

AMC10 2002 A · Q7

AMC10 2002 A · Q7. It mainly tests Ratios & proportions, Circle theorems.

If an arc of 45° on circle A has the same length as an arc of 30° on circle B, then the ratio of the area of circle A to the area of circle B is

如果圆A上45°的弧长与圆B上30°的弧长相等，则圆A的面积与圆B的面积之比是

(A) 4/9 4/9

(B) 2/3 2/3

(D) 3/2 3/2

(E) 9/4 9/4

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) Let $C_A = 2\pi R_A$ be the circumference of circle $A$, let $C_B = 2\pi R_B$ be the circumference of circle $B$, and let $L$ the common length of the two arcs. Then $\dfrac{45}{360}C_A = L = \dfrac{30}{360}C_B.$ Therefore $\dfrac{C_A}{C_B}=\dfrac{2}{3}\ \ \text{so}\ \ \dfrac{2}{3}=\dfrac{2\pi R_A}{2\pi R_B}=\dfrac{R_A}{R_B}.$ Thus, the ratio of the areas is $\dfrac{\text{Area of Circle }(A)}{\text{Area of Circle }(B)}=\dfrac{\pi R_A^2}{\pi R_B^2}=\left(\dfrac{R_A}{R_B}\right)^2=\dfrac{4}{9}.$

（A）设 $C_A = 2\pi R_A$ 为圆 $A$ 的周长，$C_B = 2\pi R_B$ 为圆 $B$ 的周长，$L$ 为两段弧的公共弧长。则 $\dfrac{45}{360}C_A = L = \dfrac{30}{360}C_B.$ 因此 $\dfrac{C_A}{C_B}=\dfrac{2}{3}\ \ \text{所以}\ \ \dfrac{2}{3}=\dfrac{2\pi R_A}{2\pi R_B}=\dfrac{R_A}{R_B}.$ 因此，面积之比为 $\dfrac{\text{圆}(A)\text{的面积}}{\text{圆}(B)\text{的面积}}=\dfrac{\pi R_A^2}{\pi R_B^2}=\left(\dfrac{R_A}{R_B}\right)^2=\dfrac{4}{9}.$

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