2021 AMC 12A Spring Problem 11

Question

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}

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

Reflect the start across the

y

-axis and the end across the

x

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

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.

Problem 11 in other years