Question

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 $3$ - $4$ - $5$ 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 $B?$

$\dfrac{9}{25}$

$\dfrac{1}{9}$

$\dfrac{1}{5}$

$\dfrac{25}{169}$

$\dfrac{4}{25}$

答案：D

知识点：外接圆、外心与外接圆半径直角三角形面积比

难度评级：1080

解答：

直角三角形的斜边是外接圆的直径。因此圆 $A$ 的直径为 $5$ ，圆 $B$ 的直径为 $13$ 。面积之比等于直径平方之比： $\left(\frac{5}{13}\right)^2 = \frac{25}{169}$ 。所以正确答案是 D。

The hypotenuse of a right triangle is a diameter of the circle around it. So circle $A$ has diameter $5$ and circle $B$ has diameter $13.$ Areas scale as the square of that, giving $\left(\frac{5}{13}\right)^2 = \frac{25}{169}.$ Thus, D is the correct answer.

2023 AMC 10B 第 3 题

答案：D

难度评级：1080

解答：

其他年份的第 3 题