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. | ||
| Line 21: | Line 20: | ||
draw(shift((4*i-1,0)) * Qp); | draw(shift((4*i-1,0)) * Qp); | ||
} | } | ||
| − | draw((-1,0)--(18.5,0 | + | 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> | ||
| − | ==Solution== | + | == Solution == |
| + | 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== | ||
| + | |||
| + | https://youtu.be/jOaLKb9Oack | ||
| − | + | 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.
Contents
Problem
The figure below shows line 
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]](http://latex.artofproblemsolving.com/c/6/6/c66986a019a481aa0dad6c2a690111ffd5a52014.png)
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
- some translation in the direction parallel to line
- the reflection across line
- some reflection across a line perpendicular to line
Solution
Statement 
is true. A 
rotation about the point half way between an up-facing square and a down-facing square will yield the same figure.
Statement 
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 
is false. A reflection across line will change the up-facing squares to down-facing squares and vice versa.
Finally, statement 
is also false because it will cause the diagonal lines extending from the squares to switch direction.
Thus, 
statements are true.
Video Solution 1
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. 
