AMC10 2023 B

AMC10 2023 B · Q3

AMC10 2023 B · Q3. It mainly tests Pythagorean theorem, Circle theorems.

A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?

一个 $3-4-5$ 直角三角形内接于圆 $A$，一个 $5-12-13$ 直角三角形内接于圆 $B$。圆 $A$ 的面积与圆 $B$ 的面积之比是多少？

(A) \frac{9}{25} \frac{9}{25}

(B) \frac{1}{9} \frac{1}{9}

(D) \frac{25}{169} \frac{25}{169}

(E) \frac{4}{25} \frac{4}{25}

Answer

Correct choice: (D)

正确答案：（D）

Solution

Because the triangles are right triangles, we know the hypotenuses are diameters of circles $A$ and $B$. Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply $\pi$ to get $6.25\pi$ and $42.25\pi$ as the areas of the circles. Multiply 4 on both numbers to get $25\pi$ and $169\pi$. Cancel out the $\pi$, and lastly, divide, to get your answer $=\boxed{\textbf{(D) }\frac{25}{169}}.$

因为这些是直角三角形，我们知道斜边是圆 $A$ 和 $B$ 的直径。因此，它们的半径分别是2.5和6.5。将这两个数字平方并乘以 $\pi$，得到圆的面积 $6.25\pi$ 和 $42.25\pi$。将两个数字都乘以4，得到 $25\pi$ 和 $169\pi$。消去 $\pi$，最后相除，得到答案 $=\boxed{\textbf{(D) }\frac{25}{169}}$。

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