Difference between revisions of "2021 AMC 12A Problems/Problem 11"

Problem

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?

$\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5$

Diagram

Graph in Desmos: https://www.desmos.com/calculator/bsiulzrjrn

~MRENTHUSIASM

Solution 1 (Condensed Explanation of Reflections)

Every time the laser bounces off a wall, instead we can imagine it going straight by reflecting it about the wall. Thus, the laser starts at $(3, 5)$ and ends at $(-7, -5)$, so the path's length is $\sqrt{10^2+10^2}=\boxed{\textbf{(C) }10\sqrt2}$.

~JHawk0224

Solution 2 (Detailed Explanation of Reflections)

Let $A=(3,5), D=(7,5), B$ be the point where the beam hits and bounces off the $y$-axis, and $C$ be the point where the beam hits and bounces off the $x$-axis.

First, reflecting $\overline{BC}$ about the $y$-axis gives $\overline{BC'}.$ Then, reflecting $\overline{CD}$ about the $y$-axis gives $\overline{C'D'}.$ Finally, reflecting $\overline{C'D'}$ about the $x$-axis gives $\overline{C'D''},$ as shown below.

Graph in Desmos: https://www.desmos.com/calculator/lxjt0ewbou

It follows that $D''=(-7,-5).$ The total distance that the beam will travel is \begin{align*} AB+BC+CD&=AB+BC'+C'D' \\ &=AB+BC'+C'D'' \\ &=AD'' \\ &=\sqrt{((3-(-7))^2+(5-(-5))^2} \\ &=\sqrt{200} \\ &=\boxed{\textbf{(C) }10\sqrt2}. \end{align*} ~MRENTHUSIASM

Solution 3 (Slopes and Parallelogram)

Define points $A,B,C,$ and $D$ as Solution 2 does.

When a straight line hits and bounces off a coordinate axis at point $P,$ the ray entering $P$ and the ray leaving $P$ have negative slopes. Let $\ell$ be the line containing $P$ and perpendicular to that coordinate axis. Geometrically, these two rays coincide when reflected about $\boldsymbol{\ell},$ creating line symmetry. Let the slope of $\overline{AB}$ be $m.$ It follows that the slope of $\overline{BC}$ is $-m,$ and the slope of $\overline{CD}$ is $m.$ Here, we conclude that $\overline{AB}\parallel\overline{CD}.$

Next, we locate $E$ on $\overline{CD}$ such that $\overline{BE}\parallel\overline{AD},$ from which $ABED$ is a parallelogram, as shown below.

Graph in Desmos: https://www.desmos.com/calculator/lgfiiqgqc2

Let $B=(0,b).$ In parallelogram $ABED,$ we get $E=(4,b).$ By symmetry, we obtain $C=(2,0).$

Applying the slope formula to $\overline{AB}$ and $\overline{DC}$ gives $$m=\frac{5-b}{3-0}=\frac{5-0}{7-2}.$$ Equating the last two expressions gives $b=2.$

By the Distance Formula, we have $AB=3\sqrt2,BC=2\sqrt2,$ and $CD=5\sqrt2.$ The total distance that the beam will travel is $$AB+BC+CD=\boxed{\textbf{(C) }10\sqrt2}.$$ ~MRENTHUSIASM

Solution 4 (Answer Choices and Educated Guesses)

Define points $A,B,C,$ and $D$ as Solution 2 does.

Since choices $\textbf{(B)}, \textbf{(C)},$ and $\textbf{(D)}$ all involve $\sqrt2,$ we suspect that one of them is the correct answer. We take a guess in faith that $\overline{AB},\overline{BC},$ and $\overline{CD}$ all form $45^\circ$ angles with the coordinate axes, from which $B=(0,2)$ and $C=(2,0).$ The given condition $D=(7,5)$ verifies our guess. Following the last paragraph of Solution 3 gives the answer $\boxed{\textbf{(C) }10\sqrt2}.$

~MRENTHUSIASM

~ pi_is_3.14

~IceMatrix