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Difference between revisions of "2021 Fall AMC 10B Problems/Problem 17"

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Now, we note that <math>(4,1)</math> is a 90 degree rotation clockwise of <math>(-1,4)</math> about the origin, which is also where <math>l</math> and <math>m</math> intersect. So <math>m</math> is a 45 degree rotation of <math>l</math> about the origin clockwise.  
 
Now, we note that <math>(4,1)</math> is a 90 degree rotation clockwise of <math>(-1,4)</math> about the origin, which is also where <math>l</math> and <math>m</math> intersect. So <math>m</math> is a 45 degree rotation of <math>l</math> about the origin clockwise.  
  
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To rotate <math>l</math> 90 degrees clockwise, we build a square with adjacent vertices <math>(0,0)</math> and <math>(1,5)</math>. The other two vertices are at <math>(5,-1)</math> and <math>(6,4)</math>. The center of the square is at <math>(3,2)</math>, which is the midpoint of <math>(1,5)</math> and <math>(5,-1)</math>. The line passes through <math>(0,0)</math> and <math>(3,2)</math>. Thus the line is <math>y = \frac{2}{3} x</math>. The answer is '''(D)''' <math>\boxed{3y - 2x = 0}</math>.
+
To rotate <math>l</math> 90 degrees clockwise, we build a square with adjacent vertices <math>(0,0)</math> and <math>(1,5)</math>. The other two vertices are at <math>(5,-1)</math> and <math>(6,4)</math>. The center of the square is at <math>(3,2)</math>, which is the midpoint of <math>(1,5)</math> and <math>(5,-1)</math>. The line <math>m</math> passes through the origin and the center of the square we built, namely at <math>(0,0)</math> and <math>(3,2)</math>. Thus the line is <math>y = \frac{2}{3} x</math>. The answer is '''(D)''' <math>\boxed{3y - 2x = 0}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=18|num-b=16}}
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=18|num-b=16}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 23:39, 23 November 2021

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) $\boxed{3y - 2x = 0}$.

See Also

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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