AMC12 2002 A

AMC12 2002 A · Q7

AMC12 2002 A · Q7. It mainly tests Circle theorems.

A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?

圆 A 上的一个 $45^\circ$ 弧长与圆 B 上的一个 $30^\circ$ 弧长相等。圆 A 的面积与圆 B 的面积之比是多少？

(A) $\frac{4}{9}$ $\frac{4}{9}$

(B) $\frac{2}{3}$ $\frac{2}{3}$

(D) $\frac{3}{2}$ $\frac{3}{2}$

(E) $\frac{9}{4}$ $\frac{9}{4}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let $r_1$ and $r_2$ be the radii of circles $A$ and$B$, respectively. It is well known that in a circle with radius $r$, a subtended arc opposite an angle of $\theta$ degrees has length $\frac{\theta}{360} \cdot 2\pi r$. Using that here, the arc of circle A has length $\frac{45}{360}\cdot2\pi{r_1}=\frac{r_1\pi}{4}$. The arc of circle B has length $\frac{30}{360} \cdot 2\pi{r_2}=\frac{r_2\pi}{6}$. We know that they are equal, so $\frac{r_1\pi}{4}=\frac{r_2\pi}{6}$, so we multiply through and simplify to get $\frac{r_1}{r_2}=\frac{2}{3}$. As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is $\boxed{\textbf{(A) } 4/9}$.

设圆 $A$ 和圆 $B$ 的半径分别为 $r_1$ 和 $r_2$。众所周知，在半径为 $r$ 的圆中，对应圆心角为 $\theta$ 度的弧长为 $\frac{\theta}{360} \cdot 2\pi r$。因此，圆 A 的弧长为 $\frac{45}{360}\cdot2\pi{r_1}=\frac{r_1\pi}{4}$，圆 B 的弧长为 $\frac{30}{360} \cdot 2\pi{r_2}=\frac{r_2\pi}{6}$。由题意两者相等，故 $\frac{r_1\pi}{4}=\frac{r_2\pi}{6}$，化简得 $\frac{r_1}{r_2}=\frac{2}{3}$。由于所有圆都相似，面积之比等于半径之比的平方，所以答案为 $\boxed{\textbf{(A) } 4/9}$。

Topics

GE.006 Circle theorems