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Difference between revisions of "2019 AMC 10A Problems/Problem 8"

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{{duplicate|[[2019 AMC 10A Problems|2019 AMC 10A #8]] and [[2019 AMC 12A Problems|2019 AMC 12A #6]]}}
 
{{duplicate|[[2019 AMC 10A Problems|2019 AMC 10A #8]] and [[2019 AMC 12A Problems|2019 AMC 12A #6]]}}
  
−
==Problem==
+
== Problem ==
−
 
 
 
The figure below shows line <math>\ell</math> with a regular, infinite, recurring pattern of squares and line segments.
 
The figure below shows line <math>\ell</math> with a regular, infinite, recurring pattern of squares and line segments.
  
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draw(shift((4*i-1,0)) * Qp);
 
draw(shift((4*i-1,0)) * Qp);
 
}
 
}
−
draw((-1,0)--(18.5,0),Arrows(TeXHead));
+
draw((-1,0)--(18.5,0));
 
</asy>
 
</asy>
 +
 
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
 
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
 
*some rotation around a point of line <math>\ell</math>
 
*some rotation around a point of line <math>\ell</math>
Line 28: Line 28:
 
*the reflection across line <math>\ell</math>
 
*the reflection across line <math>\ell</math>
 
*some reflection across a line perpendicular to line <math>\ell</math>
 
*some reflection across a line perpendicular to line <math>\ell</math>
 +
 
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math>
 
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math>
  
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==Solution==
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== Solution ==
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Statement <math>1</math> is true. A <math>180^{\circ}</math> rotation about the point in between an up-facing square and a down-facing square will yield the same figure. Statement <math>2</math> is also true. A translation to the left or right will place the image onto itself when the figures above and below the line realign (the figure goes on infinitely in both directions). Statement <math>3</math> is false. A reflection across line <math>\ell</math> will change the up-facing squares to down-facing squares and vice versa. Finally, statement <math>4</math> is also false because it will cause the diagonal lines extending from the squares to switch direction. Thus, only <math>\fbox {\textbf{(C)}  2 }</math> statements are true.
+
Statement <math>1</math> is true. A <math>180^{\circ}</math> rotation about the point half way between an up-facing square and a down-facing square will yield the same figure.  
 +
 
 +
Statement <math>2</math> is also true. A translation to the left or right will place the image onto itself when the figures above and below the line realign (the figure goes on infinitely in both directions).  
 +
 
 +
Statement <math>3</math> is false. A reflection across line <math>\ell</math> will change the up-facing squares to down-facing squares and vice versa.  
 +
 
 +
Finally, statement <math>4</math> is also false because it will cause the diagonal lines extending from the squares to switch direction.  
 +
 
 +
Thus, <math>\fbox {\textbf{(C)}  2}</math> statements are true.
 +
 
 +
==Video Solution 1==
  
−
- Violin God
+
https://youtu.be/jOaLKb9Oack
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- Abhijeet Saha 2.0
 
  
−
==See Also==
+
Education, the Study of Everything
  
 +
== See Also ==
 
{{AMC10 box|year=2019|ab=A|num-b=7|num-a=9}}
 
{{AMC10 box|year=2019|ab=A|num-b=7|num-a=9}}
 
{{AMC12 box|year=2019|ab=A|num-b=5|num-a=7}}
 
{{AMC12 box|year=2019|ab=A|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:05, 16 July 2024

The following problem is from both the 2019 AMC 10A #8 and 2019 AMC 12A #6, so both problems redirect to this page.

Problem

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0)); [/asy]

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

  • some rotation around a point of line $\ell$
  • some translation in the direction parallel to line $\ell$
  • the reflection across line $\ell$
  • some reflection across a line perpendicular to line $\ell$

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Solution

Statement $1$ is true. A $180^{\circ}$ rotation about the point half way between an up-facing square and a down-facing square will yield the same figure.

Statement $2$ is also true. A translation to the left or right will place the image onto itself when the figures above and below the line realign (the figure goes on infinitely in both directions).

Statement $3$ is false. A reflection across line $\ell$ will change the up-facing squares to down-facing squares and vice versa.

Finally, statement $4$ is also false because it will cause the diagonal lines extending from the squares to switch direction.

Thus, $\fbox {\textbf{(C)} 2}$ statements are true.

Video Solution 1

https://youtu.be/jOaLKb9Oack

Education, the Study of Everything

See Also

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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
2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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