2021 AMC 12A Spring 第 11 题

Question

一个激光器放在点

(3, 5)

。激光束沿直线传播。Larry 希望光束先击中

y

-轴并反射，再击中

x

-轴并反射，最后击中点

(7, 5)

。光束沿这条路径传播的总距离是多少？

A laser is placed at the point

(3, 5).

The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the

y

-axis, then hit and bounce off the

x

-axis, then hit the point

(7, 5).

What is the total distance the beam will travel along this path?

2\sqrt{10}

5\sqrt{2}

10\sqrt{2}

15\sqrt{2}

10\sqrt{5}

从一条直线反射，可以通过把一个端点关于这条直线反射来把路径拉直

A bounce off a line means reflecting one endpoint across that line straightens the path

将起点关于

y

-轴反射，并将终点关于

x

-轴反射，然后求两个像点之间的直线距离

Reflect the start across the

y

-axis and the end across the

x

-axis, then take the straight-line distance between the images

每次反射都可以把路径展开成一条直线。将起点

(3, 5)

关于

y

-轴反射到

(-3, 5)

，并将目标点

(7, 5)

关于

x

-轴反射到

(7, -5)

。总传播距离等于这两个像点之间的直线距离：

\begin{aligned} &\sqrt{(-3 - 7)^2 + (5 - (-5))^2} \\ &= \sqrt{100 + 100} = 10\sqrt{2} \end{aligned}\text{。}

因此，正确答案是 C。

Reflecting the path at each bounce turns it into a single straight segment. Reflect the start

(3, 5)

across the

y

-axis to

(-3, 5),

and reflect the target

(7, 5)

across the

x

-axis to

(7, -5).

The total travel distance equals the straight-line distance between these two images:

\begin{aligned} &\sqrt{(-3 - 7)^2 + (5 - (-5))^2} \\ &= \sqrt{100 + 100} = 10\sqrt{2}. \end{aligned}

Thus, the correct answer is C.

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