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?

$2\sqrt{10}$

$5\sqrt{2}$

$10\sqrt{2}$

$15\sqrt{2}$

$10\sqrt{5}$

Solution:

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: $\sqrt{(-3 - 7)^2 + (5 - (-5))^2} = \sqrt{100 + 100} = 10\sqrt{2}.$

Thus, the correct answer is C.

2021 AMC 12A Spring Problem 11

Solution:

Problem 11 in Other Years