2021 Fall AMC 10B Problems/Problem 17

Problem

Distinct lines $\ell$ and $m$ lie in the $xy$ -plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$ , and then $P'$ is reflected about line $m$ to point $P''$ . The equation of line $\ell$ is $5x - y = 0$ , and the coordinates of $P''$ are $(4,1)$ . What is the equation of line $m?$

$(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0$

Solutions

Solution 1

It is well known that the composition of 2 reflections , one after another, about two lines $l$ and $m$ , respectively, that meet at an angle $\theta$ is a rotation by $2\theta$ around the intersection of $l$ and $m$ .

Now, we note that $(4,1)$ is a 90 degree rotation clockwise of $(-1,4)$ about the origin, which is also where $l$ and $m$ intersect. So $m$ is a 45 degree rotation of $l$ about the origin clockwise.

To rotate $l$ 90 degrees clockwise, we build a square with adjacent vertices $(0,0)$ and $(1,5)$ . The other two vertices are at $(5,-1)$ and $(6,4)$ . The center of the square is at $(3,2)$ , which is the midpoint of $(1,5)$ and $(5,-1)$ . The line $m$ passes through the origin and the center of the square we built, namely at $(0,0)$ and $(3,2)$ . Thus the line is $y = \frac{2}{3} x$ . The answer is (D) $\boxed{3y - 2x = 0}$ .

~hurdler

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2021 Fall AMC 10B Problems/Problem 17

Contents

Problem

Solutions

Solution 1

See Also